The document discusses confidence intervals (CIs) and how to calculate them. Some key points:
- CIs provide a range of values that are likely to contain the true population parameter, with a specified level of confidence (usually 95%).
- The width of a CI depends on the sample size, population variation, and confidence level - larger samples and lower variation yield narrower CIs.
- CIs can be calculated for means, differences of means, proportions, and differences of proportions using formulas involving standard errors, t or Z distributions, and sample statistics. Sample size and normality assumptions affect choice of distributions.
Inferential statistics are used to draw conclusions about populations based on samples. The two primary inferential methods are estimation and hypothesis testing. Estimation involves using sample statistics to estimate unknown population parameters, such as means or proportions. Interval estimation provides a range of plausible values for the population parameter based on the sample data and a level of confidence, such as a 95% confidence interval. The width of the confidence interval depends on factors like the sample size, standard deviation, and desired confidence level.
What is statistical analysis? It's the science of collecting, exploring and presenting large amounts of data to discover underlying patterns and trends. Statistics are applied every day – in research, industry and government – to become more scientific about decisions that need to be made.
This document discusses confidence intervals for estimating population means from sample data. It begins by explaining how to calculate point estimates and confidence intervals when the sample size is large (n ≥ 30) using the normal distribution. It then covers calculating confidence intervals when the sample size is small (n < 30) using the t-distribution. The key steps covered are determining the appropriate distribution to use based on sample size and knowledge of the population standard deviation, finding the critical values and margin of error, and calculating the confidence interval. Examples are provided to demonstrate how to construct confidence intervals in different situations.
This document provides solutions to statistical estimation and confidence interval problems. It defines key statistical concepts like confidence level, margin of error, and t and chi-square distributions. Several multi-part problems are solved involving determining sample sizes needed, calculating point estimates and confidence intervals for means, proportions, variances and standard deviations using the relevant statistical formulas and distributions.
This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
Estimating population values ppt @ bec domsBabasab Patil
This document discusses confidence intervals for estimating population parameters. It covers confidence intervals for the mean when the population standard deviation is known and unknown, as well as confidence intervals for the population proportion. Key points include:
- A confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic.
- The margin of error and confidence level affect the width of a confidence interval.
- The t-distribution is used instead of the normal when the population standard deviation is unknown.
- Sample size formulas allow determining the required sample size to estimate a population parameter within a specified margin of error and confidence level.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
This document provides an overview of statistical estimation and inference. It discusses point estimation, which provides a single value to estimate an unknown population parameter, and interval estimation, which gives a range of plausible values for the parameter. The key aspects of interval estimation are confidence intervals, which provide a probability statement about where the true population parameter lies. The document also covers important concepts like sampling distributions, the central limit theorem, and factors that influence the width of a confidence interval like sample size. Examples are provided to demonstrate calculating point estimates, confidence intervals, and dealing with independent samples.
Inferential statistics are used to draw conclusions about populations based on samples. The two primary inferential methods are estimation and hypothesis testing. Estimation involves using sample statistics to estimate unknown population parameters, such as means or proportions. Interval estimation provides a range of plausible values for the population parameter based on the sample data and a level of confidence, such as a 95% confidence interval. The width of the confidence interval depends on factors like the sample size, standard deviation, and desired confidence level.
What is statistical analysis? It's the science of collecting, exploring and presenting large amounts of data to discover underlying patterns and trends. Statistics are applied every day – in research, industry and government – to become more scientific about decisions that need to be made.
This document discusses confidence intervals for estimating population means from sample data. It begins by explaining how to calculate point estimates and confidence intervals when the sample size is large (n ≥ 30) using the normal distribution. It then covers calculating confidence intervals when the sample size is small (n < 30) using the t-distribution. The key steps covered are determining the appropriate distribution to use based on sample size and knowledge of the population standard deviation, finding the critical values and margin of error, and calculating the confidence interval. Examples are provided to demonstrate how to construct confidence intervals in different situations.
This document provides solutions to statistical estimation and confidence interval problems. It defines key statistical concepts like confidence level, margin of error, and t and chi-square distributions. Several multi-part problems are solved involving determining sample sizes needed, calculating point estimates and confidence intervals for means, proportions, variances and standard deviations using the relevant statistical formulas and distributions.
This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
Estimating population values ppt @ bec domsBabasab Patil
This document discusses confidence intervals for estimating population parameters. It covers confidence intervals for the mean when the population standard deviation is known and unknown, as well as confidence intervals for the population proportion. Key points include:
- A confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic.
- The margin of error and confidence level affect the width of a confidence interval.
- The t-distribution is used instead of the normal when the population standard deviation is unknown.
- Sample size formulas allow determining the required sample size to estimate a population parameter within a specified margin of error and confidence level.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
This document provides an overview of statistical estimation and inference. It discusses point estimation, which provides a single value to estimate an unknown population parameter, and interval estimation, which gives a range of plausible values for the parameter. The key aspects of interval estimation are confidence intervals, which provide a probability statement about where the true population parameter lies. The document also covers important concepts like sampling distributions, the central limit theorem, and factors that influence the width of a confidence interval like sample size. Examples are provided to demonstrate calculating point estimates, confidence intervals, and dealing with independent samples.
This document discusses confidence intervals for population means and proportions. It explains how to construct confidence intervals using the normal distribution for large sample sizes (n ≥ 30) and the t-distribution for small sample sizes. Formulas are provided for calculating margin of error and determining necessary sample size. Guidelines are given for determining whether to use the normal or t-distribution based on sample size and characteristics. Confidence intervals can be constructed for variance and standard deviation using the chi-square distribution.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
- Confidence intervals provide an estimated range of values that is likely to include an unknown population parameter, such as a mean, with a specified degree of confidence.
- The margin of error depends on the sample size, standard deviation, and confidence level, with a larger sample size and smaller standard deviation yielding a smaller margin of error.
- When the sample size is small, a t-distribution rather than normal distribution is used to construct the confidence interval due to the unknown population standard deviation. The t-distribution is wider than the normal and accounts for additional uncertainty from an unknown standard deviation.
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1) The sample shows the mean weight of men is 172.55 lbs with a standard deviation of 26 lbs.
2) A 95% confidence interval for the population mean weight is estimated to be between 164.49 lbs and 180.61 lbs.
3) This suggests that the outdated estimate of 166.3 lbs used for safety capacities is likely an underestimate, and updating to the point estimate of 172.55 lbs could help prevent overloading issues.
Inferential statistics are often used to compare the differences between the treatment groups. Inferential statistics use measurements from the sample of subjects in the experiment to compare the treatment groups and make generalizations about the larger population of subjects.
We review important concepts from Chapters 1-5 of the statistics textbook.
1) Descriptive statistics summarize sample data, while inferential statistics make predictions about populations from samples.
2) Variables can be categorical (nominal, ordinal) or quantitative (continuous, discrete), which affects analysis methods.
3) Random sampling and random assignment in experiments reduce bias to obtain reliable data.
4) Probability distributions, the normal distribution, and the Central Limit Theorem are important concepts for statistical inference.
This document provides an overview of confidence intervals. It defines key terms like statistical inference, confidence level, and margin of error. It explains how to construct confidence intervals for means using the z-distribution when the population standard deviation is known, and using the t-distribution when it is unknown. It also covers how to estimate population proportions using the normal distribution. Examples are provided to demonstrate how to use the PANIC method to set up and calculate confidence intervals.
This document discusses confidence intervals, which are interval estimates of population parameters that indicate the reliability of sample estimates. The document defines confidence intervals and explains how they are constructed. It also discusses point estimates versus interval estimates and describes how to calculate confidence intervals for means, proportions, and when the population standard deviation is unknown using the t-distribution. Examples are provided to illustrate how to construct confidence intervals in different situations.
This document provides an overview of statistical confidence intervals. It defines key terms like point estimates, confidence levels, and confidence intervals. The document explains how to calculate confidence intervals for a population mean when the population standard deviation is known, using a point estimate, standard error, and critical z-value. It emphasizes that a confidence interval provides more information than a point estimate alone by indicating the range of uncertainty around the estimated value. Examples are given to demonstrate how to interpret a 95% confidence interval.
Ch3_Statistical Analysis and Random Error Estimation.pdfVamshi962726
Here are the steps to solve this example:
(a) Compute the sample statistics:
Mean (x̅) = (Σxi)/n = (56.13)/10 = 5.613 cm
Standard deviation (s) = √[(Σ(xi - x̅)2)/(n-1)] = 0.6266 cm
(b) The interval over which 95% of measurements should lie is:
x̅ ± t0.025,9s = 5.613 ± 2.262(0.6266) = 5.613 ± 1.417 cm
(c) The estimated true mean value at 95% probability is:
μx = x
This document discusses statistical confidence interval estimation. It covers:
1) Confidence interval estimation for the mean when the population standard deviation is known and unknown.
2) Confidence interval estimation for the proportion.
3) Factors that affect confidence interval width like data variation, sample size, and confidence level.
4) How to estimate sample sizes needed to estimate a population mean or proportion within a given level of precision and confidence.
This document discusses statistical concepts related to descriptive and inferential statistics including estimation and significance testing. It explains point estimation and interval estimation. Point estimation provides a single value estimate of a population parameter based on a sample statistic. Interval estimation provides a range of plausible values for the population parameter with a stated probability based on a sample. The document provides examples of calculating confidence intervals and explains how confidence levels and sample sizes impact the width of confidence intervals.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
The document discusses estimation and different types of estimates used to estimate population parameters based on sample data. Point estimates provide a single value while interval estimates provide a range of values. Good estimators are unbiased, efficient, and consistent. Common point estimators are the sample mean and sample standard deviation. Interval estimates use the point estimate plus/minus a margin of error calculated from the standard error. Confidence intervals provide a probability that the population parameter lies within the interval estimate.
The document discusses key concepts in statistics related to populations, samples, and sampling distributions. Some main points:
- We collect sample data to make inferences about unknown population parameters. Samples should be representative of the overall population.
- The sampling distribution of sample means approximates a normal distribution as long as sample sizes are large. This allows us to calculate confidence intervals and test hypotheses about population means and proportions.
- Common statistical tests include z-tests and t-tests for single means and proportions using the standard error to determine confidence intervals and assess significance. These can determine if sample results align with hypothesized population values.
This document provides an introduction to applied statistics and various statistical concepts. It discusses the normal (Gaussian) distribution, standard deviation, standard error of the mean, and confidence intervals. Examples and explanations are provided for each concept. Hands-on examples for calculating these statistics in Excel, SPSS, and Prism are also presented. The document aims to explain key statistical terms and how they are applied in data analysis.
This document discusses sampling distributions and their use in making statistical inferences from data. It begins by defining key aspects of sampling distributions, including the statistic of interest (e.g. mean, proportion), random selection of samples, sample size, and population. It then generates a sampling distribution using an example of calculating the mean number of months since patients' last medical examination across different samples. The document outlines important characteristics of sampling distributions and how the central limit theorem applies. It also discusses how to construct confidence intervals and conduct hypothesis testing using sampling distributions.
This document summarizes benign prostatic hyperplasia (BPH). It discusses the pathology and pathogenesis of BPH, including that it affects glandular epithelium, stromal cells, and causes increased growth. It also covers the symptomatology, evaluation, and various treatment options for BPH including watchful waiting, medical therapy, and prostatectomies. Surgical treatments discussed are transurethral resection of the prostate (TURP), retropubic prostatectomy (RPP), and transvesical prostatectomy (TVP).
This document provides an introduction to pathology. It defines pathology as the study of disease through scientific methods and examines the mechanisms of disease from etiology to clinical manifestation. The key points are:
1. Pathology studies the etiology, pathogenesis, morphologic changes, and functional derangements that result from disease processes.
2. Diseases are examined through diagnostic techniques including histopathology, cytopathology, and biochemical/immunological testing to identify structural and molecular alterations.
3. The natural course of a disease involves stages from initial exposure through biological onset, clinical onset, potential resolution or death.
This document discusses confidence intervals for population means and proportions. It explains how to construct confidence intervals using the normal distribution for large sample sizes (n ≥ 30) and the t-distribution for small sample sizes. Formulas are provided for calculating margin of error and determining necessary sample size. Guidelines are given for determining whether to use the normal or t-distribution based on sample size and characteristics. Confidence intervals can be constructed for variance and standard deviation using the chi-square distribution.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
- Confidence intervals provide an estimated range of values that is likely to include an unknown population parameter, such as a mean, with a specified degree of confidence.
- The margin of error depends on the sample size, standard deviation, and confidence level, with a larger sample size and smaller standard deviation yielding a smaller margin of error.
- When the sample size is small, a t-distribution rather than normal distribution is used to construct the confidence interval due to the unknown population standard deviation. The t-distribution is wider than the normal and accounts for additional uncertainty from an unknown standard deviation.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
1) The sample shows the mean weight of men is 172.55 lbs with a standard deviation of 26 lbs.
2) A 95% confidence interval for the population mean weight is estimated to be between 164.49 lbs and 180.61 lbs.
3) This suggests that the outdated estimate of 166.3 lbs used for safety capacities is likely an underestimate, and updating to the point estimate of 172.55 lbs could help prevent overloading issues.
Inferential statistics are often used to compare the differences between the treatment groups. Inferential statistics use measurements from the sample of subjects in the experiment to compare the treatment groups and make generalizations about the larger population of subjects.
We review important concepts from Chapters 1-5 of the statistics textbook.
1) Descriptive statistics summarize sample data, while inferential statistics make predictions about populations from samples.
2) Variables can be categorical (nominal, ordinal) or quantitative (continuous, discrete), which affects analysis methods.
3) Random sampling and random assignment in experiments reduce bias to obtain reliable data.
4) Probability distributions, the normal distribution, and the Central Limit Theorem are important concepts for statistical inference.
This document provides an overview of confidence intervals. It defines key terms like statistical inference, confidence level, and margin of error. It explains how to construct confidence intervals for means using the z-distribution when the population standard deviation is known, and using the t-distribution when it is unknown. It also covers how to estimate population proportions using the normal distribution. Examples are provided to demonstrate how to use the PANIC method to set up and calculate confidence intervals.
This document discusses confidence intervals, which are interval estimates of population parameters that indicate the reliability of sample estimates. The document defines confidence intervals and explains how they are constructed. It also discusses point estimates versus interval estimates and describes how to calculate confidence intervals for means, proportions, and when the population standard deviation is unknown using the t-distribution. Examples are provided to illustrate how to construct confidence intervals in different situations.
This document provides an overview of statistical confidence intervals. It defines key terms like point estimates, confidence levels, and confidence intervals. The document explains how to calculate confidence intervals for a population mean when the population standard deviation is known, using a point estimate, standard error, and critical z-value. It emphasizes that a confidence interval provides more information than a point estimate alone by indicating the range of uncertainty around the estimated value. Examples are given to demonstrate how to interpret a 95% confidence interval.
Ch3_Statistical Analysis and Random Error Estimation.pdfVamshi962726
Here are the steps to solve this example:
(a) Compute the sample statistics:
Mean (x̅) = (Σxi)/n = (56.13)/10 = 5.613 cm
Standard deviation (s) = √[(Σ(xi - x̅)2)/(n-1)] = 0.6266 cm
(b) The interval over which 95% of measurements should lie is:
x̅ ± t0.025,9s = 5.613 ± 2.262(0.6266) = 5.613 ± 1.417 cm
(c) The estimated true mean value at 95% probability is:
μx = x
This document discusses statistical confidence interval estimation. It covers:
1) Confidence interval estimation for the mean when the population standard deviation is known and unknown.
2) Confidence interval estimation for the proportion.
3) Factors that affect confidence interval width like data variation, sample size, and confidence level.
4) How to estimate sample sizes needed to estimate a population mean or proportion within a given level of precision and confidence.
This document discusses statistical concepts related to descriptive and inferential statistics including estimation and significance testing. It explains point estimation and interval estimation. Point estimation provides a single value estimate of a population parameter based on a sample statistic. Interval estimation provides a range of plausible values for the population parameter with a stated probability based on a sample. The document provides examples of calculating confidence intervals and explains how confidence levels and sample sizes impact the width of confidence intervals.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
The document discusses estimation and different types of estimates used to estimate population parameters based on sample data. Point estimates provide a single value while interval estimates provide a range of values. Good estimators are unbiased, efficient, and consistent. Common point estimators are the sample mean and sample standard deviation. Interval estimates use the point estimate plus/minus a margin of error calculated from the standard error. Confidence intervals provide a probability that the population parameter lies within the interval estimate.
The document discusses key concepts in statistics related to populations, samples, and sampling distributions. Some main points:
- We collect sample data to make inferences about unknown population parameters. Samples should be representative of the overall population.
- The sampling distribution of sample means approximates a normal distribution as long as sample sizes are large. This allows us to calculate confidence intervals and test hypotheses about population means and proportions.
- Common statistical tests include z-tests and t-tests for single means and proportions using the standard error to determine confidence intervals and assess significance. These can determine if sample results align with hypothesized population values.
This document provides an introduction to applied statistics and various statistical concepts. It discusses the normal (Gaussian) distribution, standard deviation, standard error of the mean, and confidence intervals. Examples and explanations are provided for each concept. Hands-on examples for calculating these statistics in Excel, SPSS, and Prism are also presented. The document aims to explain key statistical terms and how they are applied in data analysis.
This document discusses sampling distributions and their use in making statistical inferences from data. It begins by defining key aspects of sampling distributions, including the statistic of interest (e.g. mean, proportion), random selection of samples, sample size, and population. It then generates a sampling distribution using an example of calculating the mean number of months since patients' last medical examination across different samples. The document outlines important characteristics of sampling distributions and how the central limit theorem applies. It also discusses how to construct confidence intervals and conduct hypothesis testing using sampling distributions.
This document summarizes benign prostatic hyperplasia (BPH). It discusses the pathology and pathogenesis of BPH, including that it affects glandular epithelium, stromal cells, and causes increased growth. It also covers the symptomatology, evaluation, and various treatment options for BPH including watchful waiting, medical therapy, and prostatectomies. Surgical treatments discussed are transurethral resection of the prostate (TURP), retropubic prostatectomy (RPP), and transvesical prostatectomy (TVP).
This document provides an introduction to pathology. It defines pathology as the study of disease through scientific methods and examines the mechanisms of disease from etiology to clinical manifestation. The key points are:
1. Pathology studies the etiology, pathogenesis, morphologic changes, and functional derangements that result from disease processes.
2. Diseases are examined through diagnostic techniques including histopathology, cytopathology, and biochemical/immunological testing to identify structural and molecular alterations.
3. The natural course of a disease involves stages from initial exposure through biological onset, clinical onset, potential resolution or death.
This document provides an overview of preeclampsia and eclampsia. It begins with an introduction and outlines risk factors and classifications. It then describes clinical features such as hypertension and proteinuria. The pathophysiology section explains how abnormal placentation leads to reduced blood flow and imbalance of prostaglandins. Complications are also discussed, including renal failure, pulmonary edema, and intrauterine growth restriction. The document provides information on diagnosis and management of preeclampsia and eclampsia.
This seminar presentation discusses hypersensitivity reactions, which are exaggerated or inappropriate immune responses to benign antigens. It covers the objectives, mechanisms, classification, complications, and references related to hypersensitivity reactions. There are four main types of hypersensitivity reactions: Type I involves IgE antibodies and mast cell degranulation, Type II involves antibody-mediated cell cytotoxicity, Type III involves immune complex formation and deposition, and Type IV involves T-cell mediated reactions. The presentation provides examples and details of each type of hypersensitivity reaction and their clinical implications.
This document discusses inflammation. It defines inflammation as the body's local response to injury or infection aimed at eliminating the cause of injury and initiating repair. The cardinal signs of inflammation are redness, swelling, heat, pain, and loss of function. The early response involves vasodilation and increased permeability, causing swelling. The late response involves neutrophils in acute inflammation and macrophages in chronic cases, which work to destroy pathogens and initiate healing. Understanding inflammation is important for diagnosing conditions like appendicitis and treating diseases.
This document provides an overview of hyaline membrane disease (HMD), also known as respiratory distress syndrome (RDS), for nursing students. It defines RDS as a lack of pulmonary surfactant, outlines its pathophysiology and risk factors. The document discusses the clinical presentation of RDS, including respiratory distress, radiographic findings and laboratory abnormalities. It also covers diagnosis, differential diagnoses, treatment including surfactant replacement and supportive care, complications and prevention of RDS through antenatal corticosteroids.
1. Acute inflammation is rapid in onset and short in duration, characterized by fluid and protein exudation and neutrophil accumulation. Chronic inflammation is slower in onset and longer lasting, characterized by mononuclear cell infiltration, ongoing tissue destruction, and attempts at repair through fibrosis.
2. The key features of acute inflammation are vasodilation, increased vascular permeability, and recruitment of leukocytes from the blood vessels to the site of injury. Chronic inflammation features mononuclear cell infiltration, persistent tissue damage, and attempts to repair through fibrosis and angiogenesis.
3. Granulomatous inflammation is a pattern of chronic inflammation seen with certain infections, featuring focal collections of activated macrophages that develop an epithelial-like appearance known
Cellular injury can result in adaptation, reversible injury, irreversible injury leading to necrosis or apoptosis, or intracellular accumulation. The outcome depends on the injurious agent and cell type. Adaptations include hypertrophy, hyperplasia, atrophy, and metaplasia. Reversible injury includes fatty changes and pigment accumulation. Necrosis is cell death resulting from hypoxia, free radicals, membrane damage, or calcium influx. There are several types of necrosis including coagulative, liquefactive, fat, caseous, and gangrenous. Apoptosis is programmed cell death that does not cause inflammation.
This document discusses pelvic inflammatory disease (PID) and ectopic pregnancy. It defines PID as an infection of the upper female genital tract that spreads to involve the uterus, fallopian tubes, and ovaries. Common causes are Neisseria gonorrhoeae, Chlamydia trachomatis, and bacterial vaginosis. Risk factors include multiple sexual partners and past gynecological procedures. Symptoms can range from mild to severe abdominal pain. Diagnosis involves clinical exams and tests. Complications include infertility and ectopic pregnancy. Ectopic pregnancy is defined as implantation outside the uterus, most commonly in the fallopian tube. Causes may include anatomical obstructions or abnormalities in the fallop
The document discusses acid-base balance and disturbances. It defines the two main buffer systems - metabolic (kidneys) and respiratory (lungs) - that work to maintain blood pH between 7.35-7.45. Five primary acid-base imbalances are described: metabolic acidosis, metabolic alkalosis, respiratory acidosis, respiratory alkalosis, and mixed disturbances. Diagnosis involves blood tests including arterial blood gases and electrolytes to classify the disturbance based on pH, PCO2, and bicarbonate levels. Treatment focuses on addressing the underlying cause rather than just the pH effect.
This document provides an overview of autoimmune diseases. It defines autoimmune diseases as conditions where the immune system mistakenly attacks and destroys healthy body tissue. The causes include genetic factors, environmental triggers like infections, and defects in immunologic tolerance. Some specific autoimmune diseases discussed are rheumatoid arthritis, type 1 diabetes, Hashimoto's thyroiditis, Graves' disease, myasthenia gravis, and systemic sclerosis. The mechanisms, clinical features, pathology, and treatment options are described for each condition.
Patient safety is a fundamental principle of healthcare. Adverse events may result from problems in practice, products, procedures or systems. Improving patient safety demands a complex, system-wide effort involving performance improvement, risk management, infection control, safe clinical practices, and a safe environment of care. Unsafe injections expose millions of people to infections worldwide each year. Ensuring single-use injection devices and safety boxes are available in every healthcare facility can prevent reuse and unsafe waste disposal.
The document discusses integumentary disorders and provides information on the anatomy and functions of the skin. It describes common skin conditions like eczema, acne, and psoriasis. Eczema is characterized by redness, dryness, and itching. Acne presents as inflamed papules and pustules on the face and back. Psoriasis causes thickened red patches covered with silvery scales. The document outlines signs, causes, and management approaches for various dermatological disorders and skin lesions.
A nebulizer converts liquid medication into a mist that can be inhaled directly into the lungs, allowing for rapid onset of medication effects. There are different types of nebulizers that administer medication via mouthpiece or mask. Nebulizers are commonly used to treat conditions involving airflow obstruction like asthma. Proper use involves preparing equipment and medication, positioning the patient, administering the treatment, and monitoring for side effects.
This document provides an overview of the endocrine system, including the major glands and hormones. It describes the hypothalamus and pituitary glands which regulate many other endocrine glands. Other glands covered include the thyroid, parathyroid, adrenal, pancreas, ovaries, testes, thymus, and pineal. The document outlines how to assess endocrine disorders and lists some common laboratory studies. It also provides details on diabetes mellitus, describing the main types of diabetes including type 1, type 2, and gestational diabetes.
This document provides guidance on performing a cardiac and abdominal examination. It outlines the objectives, symptoms, and physical examination techniques for assessing the cardiovascular and abdominal systems. The cardiovascular section covers inspection of the jugular veins, palpation of pulses, auscultation of heart sounds, and measurement of blood pressure. The abdominal section reviews inspection, auscultation, percussion and palpation techniques. Proper examination order and identification of normal versus abnormal findings are emphasized.
This document summarizes several endocrine system disorders including hyperthyroidism, hypothyroidism, hyperparathyroidism, hypoparathyroidism, Cushing's syndrome, Conn's syndrome, Addison's disease, and pituitary adenomas. It provides epidemiological data on certain disorders and describes associated symptoms, diagnostic evaluations, and medical management approaches. Multiple endocrine neoplasia syndromes are also briefly discussed.
This document provides guidance on effectively breaking bad news to patients. It discusses the importance of this communication skill for healthcare professionals. The document outlines best practices for setting, perception checking, invitation, knowledge sharing, exploring the patient's response, and summarizing. Key aspects include ensuring privacy, empathy, clarity, and allowing time for the patient's questions and reactions. The SPIKES protocol is presented as a framework for structuring the discussion. Examples of both best practices and things to avoid are also highlighted.
2 Assessment of patient with respiratory disorder.pptxMohammedAbdela7
This document provides guidelines for performing a physical examination of the thorax and lungs. It begins by outlining the session objectives and general examination guidelines. It then discusses pertinent history data to obtain, such as cough characteristics and sputum type/color. The physical exam involves inspection, palpation, percussion, and auscultation of the chest. Inspection evaluates breathing patterns, respiratory distress signs, and overall appearance. Palpation assesses tracheal position, chest expansion, tactile fremitus, and tenderness. Percussion and auscultation are also performed to evaluate the lungs. Proper equipment, patient positioning, and exam techniques are emphasized throughout.
This document provides an overview of critical thinking, evidence-based medicine, and how to practice evidence-based medicine. It defines critical thinking as the process of conceptualizing and evaluating information to guide beliefs and actions. Evidence-based medicine is defined as integrating the best research evidence with clinical expertise and patient values/circumstances. The history of evidence-based medicine is discussed, from Cochrane's work in the 1970s highlighting gaps between research and practice, to Guyatt coining the term "evidence-based medicine" in 1991 and Sackett explaining the combination of research, expertise, and patient factors in 1996. The five steps to practice evidence-based medicine are described as developing questions, finding evidence, appraising evidence, integrating
Cell Therapy Expansion and Challenges in Autoimmune DiseaseHealth Advances
There is increasing confidence that cell therapies will soon play a role in the treatment of autoimmune disorders, but the extent of this impact remains to be seen. Early readouts on autologous CAR-Ts in lupus are encouraging, but manufacturing and cost limitations are likely to restrict access to highly refractory patients. Allogeneic CAR-Ts have the potential to broaden access to earlier lines of treatment due to their inherent cost benefits, however they will need to demonstrate comparable or improved efficacy to established modalities.
In addition to infrastructure and capacity constraints, CAR-Ts face a very different risk-benefit dynamic in autoimmune compared to oncology, highlighting the need for tolerable therapies with low adverse event risk. CAR-NK and Treg-based therapies are also being developed in certain autoimmune disorders and may demonstrate favorable safety profiles. Several novel non-cell therapies such as bispecific antibodies, nanobodies, and RNAi drugs, may also offer future alternative competitive solutions with variable value propositions.
Widespread adoption of cell therapies will not only require strong efficacy and safety data, but also adapted pricing and access strategies. At oncology-based price points, CAR-Ts are unlikely to achieve broad market access in autoimmune disorders, with eligible patient populations that are potentially orders of magnitude greater than the number of currently addressable cancer patients. Developers have made strides towards reducing cell therapy COGS while improving manufacturing efficiency, but payors will inevitably restrict access until more sustainable pricing is achieved.
Despite these headwinds, industry leaders and investors remain confident that cell therapies are poised to address significant unmet need in patients suffering from autoimmune disorders. However, the extent of this impact on the treatment landscape remains to be seen, as the industry rapidly approaches an inflection point.
Basavarajeeyam is a Sreshta Sangraha grantha (Compiled book ), written by Neelkanta kotturu Basavaraja Virachita. It contains 25 Prakaranas, First 24 Chapters related to Rogas& 25th to Rasadravyas.
Histololgy of Female Reproductive System.pptxAyeshaZaid1
Dive into an in-depth exploration of the histological structure of female reproductive system with this comprehensive lecture. Presented by Dr. Ayesha Irfan, Assistant Professor of Anatomy, this presentation covers the Gross anatomy and functional histology of the female reproductive organs. Ideal for students, educators, and anyone interested in medical science, this lecture provides clear explanations, detailed diagrams, and valuable insights into female reproductive system. Enhance your knowledge and understanding of this essential aspect of human biology.
These lecture slides, by Dr Sidra Arshad, offer a quick overview of the physiological basis of a normal electrocardiogram.
Learning objectives:
1. Define an electrocardiogram (ECG) and electrocardiography
2. Describe how dipoles generated by the heart produce the waveforms of the ECG
3. Describe the components of a normal electrocardiogram of a typical bipolar lead (limb II)
4. Differentiate between intervals and segments
5. Enlist some common indications for obtaining an ECG
6. Describe the flow of current around the heart during the cardiac cycle
7. Discuss the placement and polarity of the leads of electrocardiograph
8. Describe the normal electrocardiograms recorded from the limb leads and explain the physiological basis of the different records that are obtained
9. Define mean electrical vector (axis) of the heart and give the normal range
10. Define the mean QRS vector
11. Describe the axes of leads (hexagonal reference system)
12. Comprehend the vectorial analysis of the normal ECG
13. Determine the mean electrical axis of the ventricular QRS and appreciate the mean axis deviation
14. Explain the concepts of current of injury, J point, and their significance
Study Resources:
1. Chapter 11, Guyton and Hall Textbook of Medical Physiology, 14th edition
2. Chapter 9, Human Physiology - From Cells to Systems, Lauralee Sherwood, 9th edition
3. Chapter 29, Ganong’s Review of Medical Physiology, 26th edition
4. Electrocardiogram, StatPearls - https://www.ncbi.nlm.nih.gov/books/NBK549803/
5. ECG in Medical Practice by ABM Abdullah, 4th edition
6. Chapter 3, Cardiology Explained, https://www.ncbi.nlm.nih.gov/books/NBK2214/
7. ECG Basics, http://www.nataliescasebook.com/tag/e-c-g-basics
NVBDCP.pptx Nation vector borne disease control programSapna Thakur
NVBDCP was launched in 2003-2004 . Vector-Borne Disease: Disease that results from an infection transmitted to humans and other animals by blood-feeding arthropods, such as mosquitoes, ticks, and fleas. Examples of vector-borne diseases include Dengue fever, West Nile Virus, Lyme disease, and malaria.
share - Lions, tigers, AI and health misinformation, oh my!.pptxTina Purnat
• Pitfalls and pivots needed to use AI effectively in public health
• Evidence-based strategies to address health misinformation effectively
• Building trust with communities online and offline
• Equipping health professionals to address questions, concerns and health misinformation
• Assessing risk and mitigating harm from adverse health narratives in communities, health workforce and health system
1. Confidence Intervals
1
• We can also put confidence intervals around other sample statistics
too (not just the sample mean)
• The CI provides the precision for the sample statistic:
100(1-α)% CI: estimate ± k x (standard error)
• So the CI equals our estimate of the sample statistic plus/minus some multiple of
the standard error
• A 95% CI means that in 95 out of 100 samples of size n, the CI would contain the
population parameter
• Also common: 90% or 99% CI’s
2. Confidence Intervals
2
𝒙
ഥ± 𝒁
𝝈
𝒏
The width of the confidence interval depends on:
• the variation in the population (i.e. the standard deviation 𝜎 which is
fixed or possibly unknown) – the more variation the wider the CI
• The sample size n – the larger the sample the narrower the CI
• The significance level – 99% CI is wider than a 95% CI because we
have to be “more sure” to capture the true value
100(1-α)% CI: estimate ± k x (standard error)
3. Confidence Intervals
3
𝑥ҧ ±
𝑍
𝜎
𝑛
𝜎
𝑛
• Why does the sample size n have to be large?
• CI’s are valid when the CLT holds
• The sample size n affects the standard error
• Remember that a fraction gets closer to its numerator
as it’s denominator gets larger
• As 𝑛 → ∞ the SE
𝜎
𝑛
gets closer and closer to 𝜎
100(1-α)% CI: estimate ± k x (standard error)
4. Calculating Confidence Intervals
4
a) Single Mean
b) Difference between means - independent samples
c) Difference between means - dependent samples
d) Single Proportion
e) Difference between proportions - independent samples
5. Calculating Confidence Intervals
5
a) CI for a Single Mean (known 𝜎)
• If the sample size is large and the population SD is
known….
• The CI formula for a single mean is:
𝑥ҧ ±
𝑍
𝜎
𝑛
𝑥ҧ is the sample mean
𝜎 is the population standard deviation
𝑛 is the sample size
𝑍 is a cutoff from the standard normal distribution
90% CI: 𝑍 = 1.64
95% CI: 𝑍 = 1.96
99% CI: 𝑍 = 2.58
6. Calculating Confidence Intervals
6
a) CI for a Single Mean (known 𝜎) EXAMPLE
• We want to estimate the age of death for the US population
• We are told the population SD is 20.2 years
• In a sample of 100 people we calculate a mean age of death of 72.1
years
• a 95% confidence interval for the mean age of death is:
𝑥ҧ ± 𝑍 𝜎
𝑛
where:
𝑥ҧ = 72.1
𝜎 = 20.2
𝑛 = 100
𝑍 =1.96
72.1 + 1.96 20.2
≈ 76.06
100
72.1 − 1.96 20.2
≈ 68.14
100
95% CI: (68.14, 76.06)
7. C.I. for the difference between
population means (normally
distributed)
7
• Known variance (2 independent samples)A
100(1‐α)% C.I. for μ1 ‐ μ2 is
𝑥
ഥ
1 − 𝑥
ഥ
2 =Zα/2
1
+ 2
𝜎 2 𝜎 2
𝑛1 𝑛2
8. Calculating Confidence Intervals
8
a) CI for a Single Mean (unknown 𝜎)
• Usually the population SD is not known
• We use the sample standard deviation 𝑠 instead
• Use t-distribution instead of standard normal
• The CI formula for a single mean is:
𝑥ҧ ±
𝑡
𝑠
𝑛
𝑥ҧ is the sample mean
𝑠 is the sample standard deviation
𝑛 is the sample size
𝑡 is a cut-off from the t-distribution with 𝑛 − 1 degrees of freedom
9. Calculating Confidence Intervals
9
Student’s t-distribution
• Similar to the normal distribution but fatter tails
• Accounts for extra uncertainly from not knowing 𝜎
• Afamily of curves determined by two parameters:
– significance level 𝛼
– degrees of freedom df
• The degrees of freedom is 𝑛 − 1 because we’ve already used one piece of
information related to the variance to estimate 𝑠
10. Calculating Confidence Intervals
10
Student’s t-distribution
• Calculate cut-off values using Stata using the command:
. display invttail(df, p)
where 𝑑𝑓 = 𝑛 − 1 and p is the area in the right tail
• For a 95% CI use 𝑑𝑓 = 𝑛 − 1 and p=0.025
• Example: if n=100 the t-value for a 95% CI would be
. display invttail(99, 0.025)
=1.984217
11. Calculating Confidence Intervals
11
a) CI for a Single Mean (unknown 𝜎) EXAMPLE
Using formula:
𝑥ҧ ±
𝑡
𝑠
𝑛
The t-value has df=n-1=462-1 and p=0.025
. display invttail(461,0.025)
1.9651232
. display 87.93723 - 1.9651232 * 16.00469 / sqrt(462)=86.473988
. display 87.93723 + 1.9651232 * 16.00469 / sqrt(462)=89.400472
The 95% CI for mean zinc is (86.5, 89.4)
n 𝑥ҧ s
13. Calculating Confidence Intervals
13
Effect of significance on width of CI:
95% CI is the default (𝛼 = 0.05):
Use level () to change sig. to 99% (𝛼 = 0.01):
The CI becomes WIDER
[86.0…[86.4… 𝑥 …89.4]…89.9]
95% 99%
14. Calculating Confidence Intervals
14
Effect of sample size on width of CI:
If we increase the sample size from 31 to 131
The standard error decreases and…
The CI becomes NARROWER n=131 n=31
[145.4…[145.6… 𝑥 …148.6]…149.8]
15. Calculating Confidence Intervals
15
Effect of SD on width of CI:
If the standard deviation increases from 6 to 10
The standard error increases and……
The CI becomes WIDER s=6 s=10
[143.9…[145.4… 𝑥 …148.8]…151.3]
16. Calculating Confidence
Intervals
Comparing Z and t:
• For large n, the t-distribution approximates the normal
• Suppose the sample mean age of death is 𝑥ҧ = 72.1 yrs and that the sample SD is equal t
o
the population SD (s = 𝜎 = 20.2 yrs)
43
CI formula: Large sample size
(n=100) 95% CI:
Small sample size (n=10)
95% CI:
Normal dist.
𝜎
𝑥ҧ ± 𝑍
𝑛
Z=1.96
72.1 ± 1.96 20.2
≈
100
(68.14, 76.06)
Z=1.96
72.1 ± 1.96 20.2
≈
10
(59.58, 84.62)
t-dist.
𝑠
𝑥ҧ ± 𝑡
𝑛
t=1.98
72.1 ± 1.98 20.2
≈
100
(68.10, 76.10)
t=2.26
72.1 ± 2.26 20.2
≈
10
(57.66, 86.54)
Similar CI (large n) t has wider CI than Z (small n)
17. Calculating Confidence
Intervals
b) CI for a difference in means (independent samples)
Suppose we have two independent groups of data and calculate a sample mean and sample
for each. The CI formula is:
17
(𝑥1 − 𝑥2) ± 𝑡 𝑝
𝑠2
1 1
+
𝑛1 𝑛2
Where:
𝑥1 and 𝑥2 are the sample means
𝑛1 and 𝑛2 are the sample sizes
𝑡 is a cut-off from the t-distribution with 𝑑𝑓 = 𝑛1 + 𝑛2 − 2
𝑝
𝑠𝑝 is the pooled variance 𝑠2 =
𝑛1−1 𝑠2+ 𝑛2−1 𝑠2
1 2
𝑛1+𝑛2−2
where 𝑠1 and 𝑠2 are sample
standard deviations
18. Calculating Confidence Intervals
18
b) CI for a difference in means (independent
samples)
Assumptions:
1. The population standard deviations are approximately equal.
We check this by comparing the sample standard deviations.
2. For small sample sizes (say n<100) the population distribution should
approximately follow the normal distribution.
This is checked by assessing the sampling distribution for normality. The
assumption is fairly robust in that the formula is valid as long as the
distribution of data in the sample is approximately mound shaped and
symmetrical.
3. The 2 groups are independent.
4. The subjects within the 2 groups are independent.
20. Calculating Confidence Intervals
20
b) CI for a difference in means (independent
samples)
ttesti assumes the sample SDs are equal by default
Use unequal option if 𝑠1 and 𝑠2 are not similar
df is
affected
21. Calculating Confidence
Intervals
21
b) CI for a difference in means (independent
samples)
Suppose we want a CI for the difference in mean height between men
and women (assuming independence and normality hold…)
ttest varname, by(groupvar) Approx.
equal SDs
22. Calculating Confidence
Intervals
22
c) CI for a difference in means (dependent
samples)
• Dependent samples occur with two groups of paired or
matched data
• Usually equal sample sizes in the 2 groups (1:1)
e.g.
– patient blood pressure before and after a treatment
– patient left leg and right leg measurements
– Two groups where pairs of people have been matched on
important demographics (age, sex, etc.)
23. Calculating Confidence Intervals
c) CI for a difference in means (dependent samples)
1. Calculate the pair differences 𝑑
e.g. for each patient, d = BP_after – BP_before
2. Find the mean 𝑑ҧ and standard deviation 𝑠𝑑 of the pair
differences
3. The CI for the mean pair differences is:
𝑑ҧ ± 𝑡
𝑠𝑑
23
𝑛
Where 𝑛 is the number of pairs and t has 𝑑𝑓 = 𝑛 − 1
24. Calculating Confidence Intervals
24
c) CI for a difference in means (dependent
samples) EXAMPLE
• The heartrates of 20 patients
before and after a treatment
• Want a 95% CI for the difference in
mean heartrate
25. Calculating Confidence Intervals
25
c) CI for a difference in means (dependent
samples) EXAMPLE
First, calculate the differences
Then use the formula 𝑑ҧ ± 𝑡
𝑠𝑑 𝑛
The 95% CI is: (1.5, 11.2)
27. Calculating Confidence
Intervals
27
𝑝Ƹ =
d) CI for a single proportion
• Suppose we have a population of subjects and some of
them have a characteristic of interest and the rest don’t
– e.g. being female, having a cancer diagnosis, survived
• We want to estimate the true proportion p who have the
characteristic of interest
• If r is the number of sample subjects that have the
characteristic and n is the sample size then the sample
proportion is:
𝑟
𝑛
28. Calculating Confidence Intervals
28
𝑆𝐸 𝑝Ƹ =
d) CI for a single proportion
• If n is large enough and p is not too extreme, then the
sampling distribution of 𝑝Ƹ is normally distributed (CLT)
• The standard error of a proportion is:
𝑝Ƹ(1 − 𝑝Ƹ)
𝑛
𝑝Ƹ ±
𝑍
• The CI formula for a single proportion is:
𝑝Ƹ(1 − 𝑝Ƹ)
𝑛
where Z is a standard normal cut-off (95% CI: Z=1.96)
29. Calculating Confidence
Intervals
29
d) CI for a single proportion
𝑝Ƹ ±
𝑍
𝑝Ƹ(1 −
𝑝
Ƹ
) 𝑛
• This formula assumes the rule of thumb:
𝒏𝒑
ෝ and 𝐧(𝟏 − 𝒑
ෝ ) must both be greater than 5
• (n must be large enough too)
• Otherwise, the formula is not valid and we’d have to use
exact binomial values instead of Z cut-offs
30. Calculating Confidence Intervals
d) CI for a single proportion EXAMPLE
• Consider the 5-yr survival for lung cancer patients (Pagano p328).
• We want to estimate the proportion p who survive 5 yrs since dx.
• In a random sample of n=52 patients only r=6 survive 5 yrs
(𝑝Ƹ=r/n=6/52~0.12).
Ƹ ≈ 45.76 so the rule of thumb holds
• Check n𝑝Ƹ ≈ 6.24 and n 1 − 𝑝
• A 95% CI for p is:
𝑝Ƹ ±
𝑍
𝑝
ෝ (
1
−
𝑝
ෝ )
𝑛
95% CI: (0.03, 0.21)
So between 3% and 21% of ptx with lung cancer survive 5 yrs after
dx. 57
31. Calculating Confidence Intervals
31
d) CI for a single proportion EXAMPLE
• In a random sample of n=52 patients only r=6 survive 5 yrs
Using Stata to find the 95% CI:
cii proportions n r
Why is this answer different to the one we calculated with the formula?
Stata uses exact binomial values instead of the normal approximation
to the binomial
32. Calculating Confidence Intervals
Similarly to the CI for a single mean, the width of the CI for
a single proportion is affected by:
32
• The sample size n
– increasing the sample size makes the CI narrower/more precise
i.e. small samples have wider CI/less precision
– The standard error decreases as n increases
• The significance level
– A 99% CI is wider than a 95% CI which is wider than a 90% CI
33. Calculating Confidence
Intervals
33
e) CI for a difference in proportions (independent
samples)
• One group has sample proportion 𝑝
ෝ 1and sample size
𝑛1
• Second group has sample proportion 𝑝
ෝ 2and sample
size
𝑛2
• The CI formula is:
𝑝
ෝ 1− 𝑝
ෝ 2
± 𝑍
𝑝
ෝ 1 1
− 𝑝
ෝ 1
𝑝
ෝ 2
1 − 𝑝
ෝ 2
+
𝑛1 𝑛2
• The RHS looks complicated but really it’s just the standard error for
𝑝
ෝ 1 − 𝑝
ෝ 2
𝑆𝐸 = 𝑣𝑎𝑟(𝑝
ෝ 1) − 𝑣𝑎𝑟(𝑝
ෝ 2)
34. Calculating Confidence Intervals
34
e)
CI for a difference in proportions (independent
samples)
𝑝
ෝ 1− 𝑝
ෝ 2
± 𝑍
𝑝
ෝ 1 1 − 𝑝
ෝ 1
𝑝
ෝ 2 1 − 𝑝
ෝ 2
+
𝑛1 𝑛2
• The formula is only valid for large samples and not too extreme
values of 𝑝1 − 𝑝2
• The rule of thumb is: if 𝑝Ƹ =
𝑟1+𝑟2
𝑛1+𝑛2
𝒏𝟏𝒑
ෝ and 𝒏𝟏(𝟏 − 𝒑
ෝ ) must both be greater than 5
and
𝒏𝟐𝒑
ෝ and 𝒏𝟐(𝟏 − 𝒑
ෝ ) must both be greater than 5
• you need to be able to check the rule of thumb, and use the CI
formula (we will calculate the CI using Stata)
35. Calculating Confidence Intervals
35
e) CI for a difference in proportions (independent samples)
EXAMPLE
Stata command:
prtesti 𝑛1 𝑟1 𝑛2 𝑟2 , count
𝑛1 = 100 𝑟1= 80
𝑛2 = 100 𝑟2 = 50
36. Calculating Confidence Intervals
36
e) CI for a difference in proportions (independent
samples) EXAMPLE
The difference in proportion of ptx who had pain relief by surgery and by meds
was between 17% and 43% (a higher % of surgery ptx had pain relief
compared to med ptx)
Difference in
sample
proportions
37. One more thing…
37
• Some Stata commands used in the lectures/tutes and
Modules are different in previous versions of Stata:
If you are using Stata 14
(the latest version) the CI
commands are:
cii means n mean sd
ci means varname
cii proportions n r
If you are using an older
version of Stata the CI
commands are:
cii n mean sd
ci varname
cii n r
53. Some Operations on
Events
1/4/2021 53
• LetAand B be two events defined on the sample space Ω.
1. Union of Two events: (AUB)
– The eventAUB consists of all outcomes inAor in B or in
bothAand B.
– The eventAUB occurs ifAoccurs, or B occurs, or bothA
and B occur.
54. 2. Intersection of Two Events: (AՈB)
– The eventAՈB consists of all outcomes in bothAand B.
– The eventAՈB occurs if bothAand B occur.
1/4/2021 54
55. 3. Complement of an Event: 𝑨 or (AC) or (A′)
– The complement of the eventAis denoted by 𝐴.
– The eventAconsists of all outcomes of Ω that are not inA.
– The event 𝐴 occurs ifAdoes not.
1/4/2021 55
56. Example: (Classical
Probability)
1/4/2021 56
• Experiment: Selecting a patient randomly from a hospital
room having six beds numbered 1, 2, 3, 4, 5, and 6.
• Define the following events:
(1) E1UE2= {1; 2, 3, 4, 6} selecting an even number or a number
less than 4:
57. 2) E1UE4= {1, 2, 3, 4, 5, 6}= Ω =selecting an even number
or an odd number.
1/4/2021 57
• It can be shown that E1UE4 = Ω where E1 and E4 are called
exhaustive events.
• The union of these events gives the whole sample space.
58. 3) E1ՈE2 ={2} = selecting an even number and a
number less than 4.
1/4/2021 58
4) E1ՈE4 =∅= selecting an even number and an odd number.
59. • E1ՈE2 =∅= In this case, E1 and E4 are called disjoint (or
mutually exclusive) events.
• These kinds of events cannot occur simultaneously (together at
the same time).
1/4/2021 59
61. • Mutually Exclusive (Disjoint) Events
– The eventsAand B are disjoint (or mutually exclusive) if
E1ՈE2 =∅
I. P(AՈB)=0
II. P(AUB)=P(A) + P(B)
1/4/2021 61
66. Marginal
Probability
1/4/2021 66
• Given some variable that can be broken down into (m) categories
designated by A1, A2, . . ., Am and another jointly occurring
variable that is broken down into (s) categories designated by B1,
B2, . . . , Bs
• The marginal probability of Ai, P(Ai), is equal to the sum of the
joint probabilities ofAi with all categories of B.
• That is
67. Example: Relative Frequency or
Empirical
1/4/2021 67
• Let us consider a bivariate table for variablesAand B.
• There are three categories for both the variables,A1,A2,andA3
forAand B1, B2, and B3 for B
Joint frequency distribution for m categories ofAand s categories of B
68. • Joint probability distribution for m categories ofAand s
categories of B
1/4/2021 68
69. • Number of elements in each cell
Probabilities of events
1/4/2021 69
70. Applications of Relative Frequency (Empirical Probability)
• Let us consider a hypothetical data on four types of diseases of
200 patients from a hospital as shown below:
1/4/2021 70
• Experiment: Selecting a patient at random and observe his/her
disease type. Total number of trials, sample size, in this case, is
n =200
Disease type A B C D Total
Number of patients 90 80 20 10 200
74. Multiplication Rules of
Probability
1/4/2021 74
Let us consider a hypothetical set of data on 600 adult males
classified by their ages and smoking habits as summarized
Consider the following event:
(B1|A2) = smokes daily given that age is between 30 and 39
75. • Two-way table displaying number of respondents by age and
smoking habit of respondents smoking habit
1/4/2021 75
78. Binomial
Distribution,…
1/4/2021 78
• example;
– if all birth records for a calendar year shows that 85.8% of
the pregnancies had delivery in week 37 or later.
– The 85.8% interpreted as the probability of a recorded birth
in week 37 or later
– If we randomly select five birth records from this
population, what is the probability that exactly three of the
records will be for full-term births?
79. Binomial Distribution,…
1/4/2021 79
• Let us designate the occurrence of a record for a full-term birth
(F) as a “success” and hasten to add that a premature birth (P)
is not a failure
• It will also be convenient to assign the number 1 to a success
and the number 0 to a failure (record of a premature birth).
80. Binomial Distribution,…
1/4/2021 80
• Suppose the five birth records selected resulted in this
sequence of full term births
PFPPF
• In coded form we would write this as
10110
P(1, 0, 1, 1, 0,) =pqppq=q2p3
81. Binomial Distribution,…
1/4/2021 81
• Three successes and two failures could occur in any one of the
following additional sequences as well:
From the addition rule we know that this
probability is equal to the sum of the
individual probabilities. In the present
example we need to sum the 10q2p3’s or,
equivalently, multiply q2p3 by 10.
82. Binomial Distribution,…
1/4/2021 82
• Answer for original question is
• Since in the population, p=0.858;
q=(1-p)=(1- 0.858)=0.142
10q2p3 =10(0.142)2(0.858)3
=10 (0.0202)(0.6316)
= 0.1276
83. Combinations
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• Acombination of n objects taken x at a time is an unordered
subset of x of the n objects
• Combination is used in large sample procedures
• The number of combinations of n objects that can be formed
by taking x of them at a time is given by
n x
𝐶 =
𝑛!
𝑥! 𝑛−𝑥 !
• where x!, read x factorial, is the product of all the whole
numbers from x down to 1. That is,
x! =x(x-1)(x-2)…(1). We note that, by definition, 0!=1.
84. Binomial Distribution,…
1/4/2021 84
• Let us return to our example in which we have a sample of
n=5 birth records and we are interested in finding the
probability that three of them will be for full-term births
5 3
𝐶 =
5! 5𝑥4𝑥3𝑥2𝑥1
3! 5−3 ! (3𝑥2𝑥1) 2𝑥1 12
= = 120
= 10
85. Binomial Distribution,…
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• In our example we let x =3, the number of successes, so that n-
x = 2, the number of failures. We then may write the
probability of obtaining exactly x successes in n trials as
f(x)= n𝑪x 𝒒𝒏−𝒙 𝒑𝒙=n𝑪x 𝒑𝒙𝒒𝒏−𝒙 , for x=0, 1, 2, …,n
• The Binomial Parameters
– binomial distribution has two parameters, n and p
– μ=np,
– σ2=np(1-p) = npq
86. Poisson Distribution
• Used to model a discrete random variable representing the number of occurrences or counts of some
random events in an interval of time or space (or some volume of matter)
• The possible values of X = x are x = 0, 1, 2, 3,…
• The discrete random variable, X, is said to have a Poisson distribution with parameter (mean) λ if the
probability
1/4/2021 86
−λ 𝑥
distribution of X is given by f(x) = 𝑒 λ
𝑥 !
87. Poisson Distribution,…
1/4/2021 87
• where e = 2.71828 (the natural number).
• λ (lambda) is the parameter of the distribution and is the
average number of occurrences of the random event in the
interval
• The Poisson Process
– The occurrences of the events are independent
– The probability of the single occurrence of the event in a
given interval is proportional to the length of the interval
88. Poisson Distribution,…
1/4/2021 88
– In any infinitesimally small portion of the interval, the probability of more
than one occurrence of the event is negligible
Example
– In a study of drug-induced anaphylaxis among patients taking rocuronium
bromide as part of their anesthesia the occurrence of anaphylaxis followed a
Poisson model with λ=12 incidents per year in Norway
89. Poisson Distribution,…
1/4/2021 89
– Find the probability that in the next year, among patients
receiving rocuronium, exactly three will experience
anaphylaxis
3!
−12 3
f(x=3) = 𝑒 12
= 0.00177
• What is the probability that at least three patients in the next
year will experience anaphylaxis if rocuronium is administered
with anesthesia?
90. Poisson
Distribution
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• Example: Suppose that the number of accidents per day in a city has
a Poisson distribution with average 2 accidents.
1. What is the probability that in a day
I. the number of accidents will be 5,
II. the number of accidents will be less than 2.
2. What is the probability that there will be six accidents in 2
days?
3. What is the probability that there will be no accidents in an
hour?
92. Probability Distributions of
Continuous data
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• A non-negative function f(x) of the continuous random
variable X if the total area bounded by its curve and the x -axis
is equal to 1 and if the subarea under the curve bounded by the
curve, the x -axis, and perpendiculars erected at any two points
a and b give the probability that X is between the points a and
b.
• Also known as probability density function
93. Probability Distributions of Continuous
data…
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Graph of a continuous distribution showing area between a
and b.
94. Normal
distribution
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• Known as the Gaussian distribution
• The normal density is given by
2𝜋
𝜎
f(x)= 1
𝑒− 𝑥−𝜇 2
/2𝜎2
, −∞ < 𝑥 < ∞;
where (e = 2.71828) and (π = 3.14159).
• The parameters of the distribution are the µ and the σ2
X ~N (μ,σ2).
95. Normal
Distribution,…
1/4/2021 95
• The density function of X, f(x), is a bell-shaped curve
– The highest point of the curve of f(x) is at the mean μ.
Hence, the mode = mean = μ.
– The curve of f(x) is symmetric about the mean μ.
– In other words, mean = mode = median
– The area under the curve is 1
96. Standard Normal
Distribution
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• the standard normal distribution with mean µ = 0 and variance
σ2 =1
• Denoted by Normal (0,1) or N(0,1).
• The standard normal random variable is denoted by Z, and we
write Z~N(0,1)
Z=𝑥−𝜇
𝜎
• The equation for the standard normal distribution is written
Z=
1
2𝜋
2
𝑒−𝑧 /2, −∞ < 𝑧 < ∞
98. Standard Normal Distribution,…
1/4/2021 98
• Z-transformation that yields a value of Z, Z=1 indicates that
the value of x used in the transformation is 1 standard
deviation above 0.
• A value of Z = -1 indicates that the value of x used in the
transformation is 1 standard deviation below 0.
99. Standard Normal Distribution,…
1/4/2021 99
• Example;
– What is the probability that a z picked at random from the
population of z’s will have a value between -2.55 and 2.55?
answer:
P(-2.55<z<2.55)=0.9946-
0.0054
=0.9892
101. Application of normal distribution
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• Normal distribution is not a law that is adhered to by all
measurable characteristics occurring in nature
• However, many of these characteristics are approximately
normally distributed
• Used to model the distribution of many variables that are of interest
• Allows us to make useful probability statements about some variables
conveniently than would be the case if some more complicated model had to be
used
102. Application,…
1/4/2021 102
• Example:
– Let us consider weight of women in reproductive age
follows a normal distribution with mean 49 kg and variance
25 kg2
a. Find the probability that a randomly chosen woman in
her reproductive age has weight less than 45 kg.
b. What is the percentage of women having weight less
than 45 kg?
c. In a population of 20,000 women of reproductive age,
how many would you expect to have weight less than
45 kg?
103. • Solution
– Here the random variable, X = weight of women in
reproductive age, population mean = 49 kg, population
variance= σ2 = 25 kg2, population standard deviation = σ =
5 kg. Hence, X~Normal (49,25).
a. The probability that a randomly chosen woman in
reproductive age has weight less than 45 kg is P(X<45)
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104. Application,…
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– The percentage of women of reproductive age who have
weight less than 45 kg is P(x<45) x100% = 0.2119 x100%
= 21.19%
• In a population of 20,000 women of reproductive age, we
would expect that the number of women with weight less than
45 kg is P(X <45)x 20,000 =0.2119 x20,000 = 4238.
105. Advantages of sampling:
105
• Feasibility: Sampling may be the only feasible method of
collecting the information.
• Reduced cost: Sampling reduces demands on resource such
as finance, personnel, and material.
• Greater accuracy: Sampling may lead to better accuracy of
collecting data
• Sampling error: Precise allowance can be made for sampling
error
• Greater speed: Data can be collected and summarized more
quickly
106. Disadvantages of sampling:
106
• There is always a sampling error.
• Sampling may create a feeling of discrimination within the
population.
• Sampling may be inadvisable where every unit in the
population is legally required to have a record.
107. Sampling technique
107
• There are two different approaches to sampling in survey research:
– Nonprobability sampling
– Probability sampling
108. Probability sampling methods
• A sample obtained in a way that every number of the
population has a known &non-zero.
– Probability of being include in the sample i.e. involves
random selection of sample
– Involves the selection of a sample from a population,
based on chance
• Probability sampling is
– More complex,
– More time-consuming
– Usually more costly than non-probability sampling.
109. EXAMPLE OF SIMPLE RANDOM SAMPLING
Age at first sex and associated factors for early sexual initiation
among students at University of AU, Central Ethiopia
– There are a total of 8, 000 students
– We want to select 700 sample students
– In this case, we assumed homogeneity with respect to age at first sex
– Their ID can be taken as frame
– Hence we can use computer generated random number to select 700
students randomly
110. Steps in systematic random sampling
1.Number the units on your frame from 1 to N (where N is the
total population size).& n=sample size
2. Determine the sampling interval (K) by dividing the number of
units in the population by the desired sample size. K=N/n k=sampling
interval=population size n=sample size
3. Draw a random number between one and K. This number is called the
random start and would be the first number included in your sample.
– Let the selected number be j
4. Select every Kth unit after that first number j, j+k, j+2k, j+3k----
-----------------j+nk
111. EXAMPLE
A systematic sample is to be selected from 1200 students of a school.
The sample size selected is 100. The sampling fraction is (skip interval)
k=1200/100=12
• The number of the first student to be included in the sample is chosen
randomly, for example by blindly 30 picking one out of twelve pieces of
paper, numbered 1- 12.
• If number 6 is picked, then every twelfth student will be included in the
sample, starting with student number 6, until 100 students are selected:
then numbers selected would be 6, 18, 30, 42, etc.
112. Stratified sampling …
The procedures are:
– Divided the total population into different homogeneous
subgroups (strata)
– Allocate sample for each strata (ni)
• Proportional allocation (ni =Ni(n/N))» Where
ni =sample for each strata
Ni=total population of each strata
n=required sample size
N=total population of the
• Disproportional (equal allocation) is some times also
possible
113. Example
• A survey is conducted on household water supply in a district comprising
20,000 households, of which 20% are urban and 80% rural
• It is suspected that in urban areas the access to safe water sources is much
more satisfactory. The total population of the district is 10, 000 (urban=4000
and rural=6000). The sample size required has been decided to be 300
• Allocate the sample proportionally for both strata?
n
urban= 4000*300/10,000=120
n
rural= 6000*300/10,000=180
114. Steps in cluster sampling
• The reference population (homogeneous) is divided into clusters.
– These clusters are often geographic units (e.g. districts, villages, etc.).
• A number of clusters are selected randomly to represent the total population,
and then all units within selected clusters are included in the sample.
• No units from non-selected clusters are included in the sample—they are
represented by those selected clusters
– This differs from stratified sampling, where some units are selected from
each group
– All the units in the selected clusters are studied
115. Example
• In a study of knowledge, attitudes, and practices related to family planning in
rural communities of a region, a list is made of all the villages.
• Using this list, a random sample of villages is chosen and all the adults in the
selected villages are interviewed
116. Multi-stage sampling
In a study of utilization of pit latrines in a district, 150 homesteads are to be
visited for interviews with family members as well as for observations on
types and cleanliness of latrines.
• The district is composed of six wards and each ward has between six and
nine villages.
• The following four stage sampling procedure could be performed:
– Select three wards out of the six by simple random sampling
– For each ward, select five villages by simple random sampling (15 villages
in total)
117. For each village select ten households. Because simply choosing
households in the center of the village would produce a biased
sample, the following systematic sampling procedure is
proposed:
– Go to the center of the village
– Choose a direction in random way
– Walk in the chosen direction and select every third or every fifth
household (depending on the size of the village) until you have the ten you
need.
118. PROBLEM
A population of cancer patients has survival standard
deviation of 43.4 months. If one wants to conduct a
study on these populations how large sample size is
needed, so that 95% of the sample mean of this size will
be within ±6 months of the population mean. Population
size is 480 patients. (85)
119. In a survey of school children to determine the population of
immunized children against polio, an investigator determined the
maximum discrepancy b/n sample and population proportion of
immunized to be 0.04, at level of confidence of 99%.further the
investigator had a previous knowledge on the prevalence among
children in a similar community to be 90% and the total
population of school children is 800.
120. The mean weight of 100 children who are 5 years old in a certain
locality is found to be 14 kg. A clinician wants to know the mean
weight of all the children in that locality with 95 % confidence
interval, if it is known that the SD for all children is 4kg
121. suppose a survey conducted on a reprehensive
sample of 900 newborn babies in A/A and it is
found that their average weight at birth is 3.5 kg
with SD of 0.5Kg. estimate the wt of newborn
babies in A/A at 95% level of confidence.
122. sample of 20 houses studied to estimate the
mean sprayable area of house for controlling of
malaria
epidemic. The result was =22.9m2, SD is
6.0m.construct CI for mean sprayable of area of
the
population with 95% confidence.
123. A random sample of 100 people shows that 25
are left-handed. Form a 95% CI for the true
proportion of
left-handers.
124.
125.
126. In a clinical trial for a new drug to treat hypertension, N1 = 50
patients were randomly assigned to receive the new drug, and N2 =
50 patients to receive a placebo. 34 of the patients receiving the drug
showed improvement, while 15 of those receiving placebo showed
improvement.
– Compute a 95% CI estimate for the difference between proportions
improved.
127. A simple random sample of 10 people from a certain
population has a mean age of 27. Can we conclude that the
mean age of the population is not 30? The variance is known
to be 20. Let CL = .95.
Data
n = 10, sample mean = 27, 2 = 20, α = 0.05
B. Assumptions
Simple random sample
Normally distributed population
128. A simple random sample of 14 people from a certain
population gives a sample mean body mass index (BMI)
of 30.5 and sd of 10.64. Can we conclude that the BMI
is not 35 at α 5%?
129. The means SUA levels on 12 individuals with Down’s
syndrome and 15 normal individuals are 4.5 and 3.4
mg/100 ml, respectively. With variances. ( 2=1,
2=1.5, respectively). Is there a difference between the
means of both groups at α 5%?
130. We wish to know if we may conclude, at the 95%
confidence level, that smokers, in general, have
greater lung damage than do non-smokers.
131. In the general population of 0 to 4-year-olds, the annual
incidence of asthma is 1.4%. If 10 cases of asthma are
observed over a single year in a sample of 500 children
whose mothers smoke, can we
conclude that this is different from the underlying
probability of p0 = 0.014 (or p=1.4%)? cl = 95%
132. Among the 225 students who ate the sandwiches, 109 became ill.
While, among the 38 students who did not eat the sandwiches, 4
became ill. Is there a significant difference between the two
groups at α
=5%