This chapter discusses methods for constructing confidence intervals for differences between population means and proportions in various sampling situations. It covers confidence intervals for the difference between two dependent or paired sample means, two independent sample means when population variances are known or unknown, and two independent population proportions. It also addresses determining required sample sizes to estimate a mean or proportion within a specified margin of error.
Chap05 continuous random variables and probability distributionsJudianto Nugroho
This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.
This chapter discusses sampling and sampling distributions. The key points are:
1) A sample is a subset of a population that is used to make inferences about the population. Sampling is important because it is less time consuming and costly than a census.
2) Descriptive statistics describe samples, while inferential statistics make conclusions about populations based on sample data. Sampling distributions show the distribution of all possible values of a statistic from samples of the same size.
3) The sampling distribution of the sample mean is normally distributed for large sample sizes due to the central limit theorem. Its mean is the population mean and its standard deviation decreases with increasing sample size. Acceptance intervals can be used to determine the range a
This document provides an overview of the binomial probability distribution, including key terminology like random experiments, outcomes, sample space, and discrete vs. continuous random variables. It defines a binomial experiment as having n repeated trials with two possible outcomes (success/failure), where the probability of success p is constant for each trial. The number of successes is a binomial random variable with a binomial probability distribution. Several examples are given to illustrate calculating probabilities of outcomes for binomial experiments involving dice rolls, patient recoveries, telephone call successes, ratios of children's sexes, and metal piston rejects. The mean, variance, and standard deviation of the binomial distribution are also defined in terms of n, p, and q.
T test, Student’s t Test, Key Takeaways, Uses of t-test / Application , Type of t-test, Type of t-test Cont.., One-tailed or two-tailed t-test, Which t-test to Use, t-test Formula, The t-score, Understanding P-values, Degrees of Freedom, How is the t-distribution table used, Example, Example Cont.., Different t-test Formulae, Different t-test Formulae Cont.., Reference.
T-distribution is the most famous theoretical probability distribution in continuous family of distributions. T distribution is used in estimation where normal distribution cannot be used to estimate population parameters. Copy the link given below and paste it in new browser window to get more information on T distribution:- http://www.transtutors.com/homework-help/statistics/t-distribution.aspx
This document discusses random sampling and sampling distributions. It covers types of random sampling including simple random sampling, stratified random sampling, cluster sampling, convenience sampling, and judgmental sampling. It then discusses sampling distributions of the mean and variance, and the central limit theorem. Key concepts covered include parameter vs statistic, constructing sampling distributions, calculating the mean and variance of sampling distributions, and applying the central limit theorem to solve problems. The document provides examples and step-by-step solutions to demonstrate these statistical concepts.
This chapter discusses hypothesis testing for the difference between two population means and two population proportions. It covers tests for:
1) Matched or dependent pairs, using a t-test and assuming normal distributions.
2) Independent populations when variances are known, using a z-test.
3) Independent populations when variances are unknown but assumed equal, using a pooled variance t-test.
4) Independent populations when variances are unknown and assumed unequal, requiring other techniques.
The document provides examples and decision rules for conducting hypothesis tests on differences between two means or proportions in various situations. Formulas for calculating test statistics like z-scores and t-statistics are presented.
Chap05 continuous random variables and probability distributionsJudianto Nugroho
This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.
This chapter discusses sampling and sampling distributions. The key points are:
1) A sample is a subset of a population that is used to make inferences about the population. Sampling is important because it is less time consuming and costly than a census.
2) Descriptive statistics describe samples, while inferential statistics make conclusions about populations based on sample data. Sampling distributions show the distribution of all possible values of a statistic from samples of the same size.
3) The sampling distribution of the sample mean is normally distributed for large sample sizes due to the central limit theorem. Its mean is the population mean and its standard deviation decreases with increasing sample size. Acceptance intervals can be used to determine the range a
This document provides an overview of the binomial probability distribution, including key terminology like random experiments, outcomes, sample space, and discrete vs. continuous random variables. It defines a binomial experiment as having n repeated trials with two possible outcomes (success/failure), where the probability of success p is constant for each trial. The number of successes is a binomial random variable with a binomial probability distribution. Several examples are given to illustrate calculating probabilities of outcomes for binomial experiments involving dice rolls, patient recoveries, telephone call successes, ratios of children's sexes, and metal piston rejects. The mean, variance, and standard deviation of the binomial distribution are also defined in terms of n, p, and q.
T test, Student’s t Test, Key Takeaways, Uses of t-test / Application , Type of t-test, Type of t-test Cont.., One-tailed or two-tailed t-test, Which t-test to Use, t-test Formula, The t-score, Understanding P-values, Degrees of Freedom, How is the t-distribution table used, Example, Example Cont.., Different t-test Formulae, Different t-test Formulae Cont.., Reference.
T-distribution is the most famous theoretical probability distribution in continuous family of distributions. T distribution is used in estimation where normal distribution cannot be used to estimate population parameters. Copy the link given below and paste it in new browser window to get more information on T distribution:- http://www.transtutors.com/homework-help/statistics/t-distribution.aspx
This document discusses random sampling and sampling distributions. It covers types of random sampling including simple random sampling, stratified random sampling, cluster sampling, convenience sampling, and judgmental sampling. It then discusses sampling distributions of the mean and variance, and the central limit theorem. Key concepts covered include parameter vs statistic, constructing sampling distributions, calculating the mean and variance of sampling distributions, and applying the central limit theorem to solve problems. The document provides examples and step-by-step solutions to demonstrate these statistical concepts.
This chapter discusses hypothesis testing for the difference between two population means and two population proportions. It covers tests for:
1) Matched or dependent pairs, using a t-test and assuming normal distributions.
2) Independent populations when variances are known, using a z-test.
3) Independent populations when variances are unknown but assumed equal, using a pooled variance t-test.
4) Independent populations when variances are unknown and assumed unequal, requiring other techniques.
The document provides examples and decision rules for conducting hypothesis tests on differences between two means or proportions in various situations. Formulas for calculating test statistics like z-scores and t-statistics are presented.
This chapter discusses time-series forecasting and index numbers. It aims to develop basic forecasting models using smoothing methods like moving averages and exponential smoothing. It also covers trend-based forecasting using linear and nonlinear regression models. Time-series data contains trend, seasonal, cyclical, and irregular components that must be accounted for. Forecasting future values involves identifying patterns in historical data and extending those patterns into the future.
This chapter discusses sampling and sampling distributions. It defines key terms like population, parameter, sample, and statistic. It also differentiates between a population and a sample. The chapter covers different sampling methods like simple random sampling, stratified random sampling, and cluster sampling. It describes the properties of the sampling distribution of the sample mean, including its expected value and standard deviation. The chapter also explains the central limit theorem.
Chap04 discrete random variables and probability distributionJudianto Nugroho
This document provides an overview of key concepts in chapter 4 of the textbook "Statistics for Business and Economics". The chapter goals are to understand mean, standard deviation, and probability distributions for discrete random variables, including the binomial, hypergeometric, and Poisson distributions. It introduces discrete random variables and probability distributions, and defines important properties like expected value, variance, and cumulative probability functions. It also covers the Bernoulli distribution as well as the characteristics and applications of the binomial distribution.
This chapter discusses confidence interval estimation. It covers constructing confidence intervals for a single population mean when the population standard deviation is known or unknown, as well as confidence intervals for a single population proportion. The chapter defines key concepts like point estimates, confidence levels, and degrees of freedom. It provides examples of how to calculate confidence intervals using the normal, t, and binomial distributions and how to interpret the resulting intervals.
Siegel-Tukey test named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to the data measured at least on an ordinal scale. It tests for the differences in scale between two groups.
The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other.
The test was published in 1980 by Sidney Siegel and John Wilder Tukey in the journal of the American Statistical Association in the article “A Non-parametric Sum Of Ranks Procedure For Relative Spread in Unpaired Samples “.
This document discusses statistical concepts related to sampling and making inferences about populations based on samples. It covers simple random sampling, point estimation, sampling distributions, and provides an example using data from St. Andrew's University applicants. Specifically, it looks at taking a census versus sampling to estimate the average SAT score and proportion wanting on-campus housing. It demonstrates how to select a simple random sample using a random number table and computer-generated random numbers in Excel. Point estimates and their sampling distributions are defined.
This document discusses simple random sampling. It begins by defining simple random sampling as selecting a sample from a population where each individual has an equal probability of being selected at each stage of sampling. It then discusses two common methods for obtaining a simple random sample: the lottery method and using a random number table. The document also explains the difference between simple random sampling with replacement (SRSWR), where selected units are replaced before subsequent selections, and simple random sampling without replacement (SRSWOR), where selected units are not replaced. It provides formulas for calculating the probability of selecting a given unit at a given draw for both SRSWR and SRSWOR.
A random variable X has a continuous uniform distribution if its probability density function f(x) is constant over the interval (α, β). The uniform distribution has a probability density function f(x) = k for α < x < β, where k is a constant, and is equal to 0 otherwise. All values from α to β are equally likely to occur, meaning the probability of X falling in any sub-interval of (α, β) is equal regardless of the interval's position within the range.
This document provides information about the binomial distribution including:
- The conditions that define a binomial experiment with parameters n, p, and q
- How to calculate binomial probabilities using the formula or tables
- How to construct a binomial distribution and graph it
- The mean, variance, and standard deviation of a binomial distribution are np, npq, and sqrt(npq) respectively
The document discusses the binomial probability distribution. It defines a binomial experiment as having n repeated trials with two possible outcomes (success/failure), a constant probability p of success on each trial, and independent trials. It provides examples of binomial experiments like coin tosses and MCQ questions. The number of successes is a binomial random variable with possible values from 0 to n. The binomial distribution gives the probability of x successes based on n, p, and the formula P(x) = (nCx) * px * q(n-x). It demonstrates calculating probabilities for different values of x.
The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, even if the population is not normally distributed. It provides the mean and standard deviation of the sampling distribution of the sample mean. The document gives the definition of the central limit theorem and provides an example of how to use it to calculate probabilities related to the sample mean of a large normally distributed population.
This chapter discusses confidence interval estimation for means and proportions. It introduces key concepts such as point estimates, confidence intervals, and confidence levels. For a mean where the population standard deviation is known, the confidence interval formula uses the normal distribution. When the standard deviation is unknown, the t-distribution is used instead. For a proportion, the confidence interval adds an allowance for uncertainty to the sample proportion. The chapter also covers determining sample sizes and interpreting confidence intervals.
1) The document discusses parametric tests and the t-test/Student's t-test. It provides examples of different types of parametric tests and explains what assumptions are made.
2) There are several types of t-tests that are used to compare means, including independent samples t-tests, paired samples t-tests, and one-sample t-tests. The t-test calculates a t-value to determine if there is a significant difference between group means.
3) The assumptions of the independent samples t-test include independent observations, normally distributed data, equal variances between groups, and random sampling. The paired t-test assumes independence of differences and a normal distribution of differences.
This chapter discusses two-sample hypothesis tests for comparing population means and proportions between two independent samples, and between two related samples. It introduces tests for comparing the means of two independent populations, two related populations, and the proportions of two independent populations. The key tests covered are the pooled variance t-test for independent samples with equal variances, separate variance t-test for independent samples with unequal variances, and the paired t-test for related samples. Examples are provided to demonstrate how to calculate the test statistic and conduct hypothesis tests to compare sample means and determine if they are statistically different. Confidence intervals for the difference between two means are also discussed.
The document discusses key probability concepts including probability, binomial distribution, normal distribution, and Poisson distribution. It provides examples of how each concept is applied in pharmaceutical research and drug development, such as calculating the probability of adverse drug events, modeling drug response rates, and analyzing the number of medication errors at a pharmacy.
The Central Limit Theorem describes how the sampling distribution of sample means approaches a normal distribution as sample size increases, even if the population is not normally distributed. Specifically, it states that the sampling distribution of sample means will be approximately normally distributed whenever the sample size is large, and the larger the sample, the better the normal approximation. The Central Limit Theorem also predicts that the mean of the sampling distribution will equal the population mean, and the standard deviation of the sampling distribution will equal the population standard deviation divided by the square root of the sample size.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
A random variable is a variable whose values are determined by the outcome of a random experiment and can be used to model probabilities. Examples of random variables include the sum of dice rolls or number of heads from coin tosses. A probability distribution assigns probabilities to each possible value of a random variable. It must satisfy the properties that probabilities are greater than or equal to 0 and sum to 1. Common probability distributions include the binomial and normal distributions.
Chi‑square Test and its Application in Hypothesis Testing
Rakesh Rana, Richa Singhal
Statistical Section, Central Council for Research in Ayurvedic Sciences, Ministry of AYUSH, GOI, New Delhi, India
This document discusses probability distributions and binomial distributions. It defines:
i) Discrete and continuous probability distributions.
ii) The binomial distribution properties including the number of trials (n), probability of success (p), and probability of failure (q).
iii) How to calculate the mean, variance, and standard deviation of a binomial distribution using the formulas: mean=np, variance=npq, and standard deviation=√(npq).
This chapter discusses additional topics related to estimation, including forming confidence intervals for differences between dependent and independent population means and proportions. It provides formulas and examples for confidence intervals when population variances are known or unknown, and when variances are assumed equal or unequal. The chapter goals are to enable readers to compute confidence intervals and determine sample sizes for a variety of estimation problems involving means, proportions, differences between populations, and variances.
This chapter discusses hypothesis testing for differences between population means and variances. It covers tests for differences between two related population means using matched pairs, differences between two independent population means when variances are known and unknown, and tests of differences between two population variances. The key test statistics are t-tests and z-tests, and the chapter presents the assumptions, test statistics, and decision rules for each hypothesis test.
This chapter discusses time-series forecasting and index numbers. It aims to develop basic forecasting models using smoothing methods like moving averages and exponential smoothing. It also covers trend-based forecasting using linear and nonlinear regression models. Time-series data contains trend, seasonal, cyclical, and irregular components that must be accounted for. Forecasting future values involves identifying patterns in historical data and extending those patterns into the future.
This chapter discusses sampling and sampling distributions. It defines key terms like population, parameter, sample, and statistic. It also differentiates between a population and a sample. The chapter covers different sampling methods like simple random sampling, stratified random sampling, and cluster sampling. It describes the properties of the sampling distribution of the sample mean, including its expected value and standard deviation. The chapter also explains the central limit theorem.
Chap04 discrete random variables and probability distributionJudianto Nugroho
This document provides an overview of key concepts in chapter 4 of the textbook "Statistics for Business and Economics". The chapter goals are to understand mean, standard deviation, and probability distributions for discrete random variables, including the binomial, hypergeometric, and Poisson distributions. It introduces discrete random variables and probability distributions, and defines important properties like expected value, variance, and cumulative probability functions. It also covers the Bernoulli distribution as well as the characteristics and applications of the binomial distribution.
This chapter discusses confidence interval estimation. It covers constructing confidence intervals for a single population mean when the population standard deviation is known or unknown, as well as confidence intervals for a single population proportion. The chapter defines key concepts like point estimates, confidence levels, and degrees of freedom. It provides examples of how to calculate confidence intervals using the normal, t, and binomial distributions and how to interpret the resulting intervals.
Siegel-Tukey test named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to the data measured at least on an ordinal scale. It tests for the differences in scale between two groups.
The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other.
The test was published in 1980 by Sidney Siegel and John Wilder Tukey in the journal of the American Statistical Association in the article “A Non-parametric Sum Of Ranks Procedure For Relative Spread in Unpaired Samples “.
This document discusses statistical concepts related to sampling and making inferences about populations based on samples. It covers simple random sampling, point estimation, sampling distributions, and provides an example using data from St. Andrew's University applicants. Specifically, it looks at taking a census versus sampling to estimate the average SAT score and proportion wanting on-campus housing. It demonstrates how to select a simple random sample using a random number table and computer-generated random numbers in Excel. Point estimates and their sampling distributions are defined.
This document discusses simple random sampling. It begins by defining simple random sampling as selecting a sample from a population where each individual has an equal probability of being selected at each stage of sampling. It then discusses two common methods for obtaining a simple random sample: the lottery method and using a random number table. The document also explains the difference between simple random sampling with replacement (SRSWR), where selected units are replaced before subsequent selections, and simple random sampling without replacement (SRSWOR), where selected units are not replaced. It provides formulas for calculating the probability of selecting a given unit at a given draw for both SRSWR and SRSWOR.
A random variable X has a continuous uniform distribution if its probability density function f(x) is constant over the interval (α, β). The uniform distribution has a probability density function f(x) = k for α < x < β, where k is a constant, and is equal to 0 otherwise. All values from α to β are equally likely to occur, meaning the probability of X falling in any sub-interval of (α, β) is equal regardless of the interval's position within the range.
This document provides information about the binomial distribution including:
- The conditions that define a binomial experiment with parameters n, p, and q
- How to calculate binomial probabilities using the formula or tables
- How to construct a binomial distribution and graph it
- The mean, variance, and standard deviation of a binomial distribution are np, npq, and sqrt(npq) respectively
The document discusses the binomial probability distribution. It defines a binomial experiment as having n repeated trials with two possible outcomes (success/failure), a constant probability p of success on each trial, and independent trials. It provides examples of binomial experiments like coin tosses and MCQ questions. The number of successes is a binomial random variable with possible values from 0 to n. The binomial distribution gives the probability of x successes based on n, p, and the formula P(x) = (nCx) * px * q(n-x). It demonstrates calculating probabilities for different values of x.
The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, even if the population is not normally distributed. It provides the mean and standard deviation of the sampling distribution of the sample mean. The document gives the definition of the central limit theorem and provides an example of how to use it to calculate probabilities related to the sample mean of a large normally distributed population.
This chapter discusses confidence interval estimation for means and proportions. It introduces key concepts such as point estimates, confidence intervals, and confidence levels. For a mean where the population standard deviation is known, the confidence interval formula uses the normal distribution. When the standard deviation is unknown, the t-distribution is used instead. For a proportion, the confidence interval adds an allowance for uncertainty to the sample proportion. The chapter also covers determining sample sizes and interpreting confidence intervals.
1) The document discusses parametric tests and the t-test/Student's t-test. It provides examples of different types of parametric tests and explains what assumptions are made.
2) There are several types of t-tests that are used to compare means, including independent samples t-tests, paired samples t-tests, and one-sample t-tests. The t-test calculates a t-value to determine if there is a significant difference between group means.
3) The assumptions of the independent samples t-test include independent observations, normally distributed data, equal variances between groups, and random sampling. The paired t-test assumes independence of differences and a normal distribution of differences.
This chapter discusses two-sample hypothesis tests for comparing population means and proportions between two independent samples, and between two related samples. It introduces tests for comparing the means of two independent populations, two related populations, and the proportions of two independent populations. The key tests covered are the pooled variance t-test for independent samples with equal variances, separate variance t-test for independent samples with unequal variances, and the paired t-test for related samples. Examples are provided to demonstrate how to calculate the test statistic and conduct hypothesis tests to compare sample means and determine if they are statistically different. Confidence intervals for the difference between two means are also discussed.
The document discusses key probability concepts including probability, binomial distribution, normal distribution, and Poisson distribution. It provides examples of how each concept is applied in pharmaceutical research and drug development, such as calculating the probability of adverse drug events, modeling drug response rates, and analyzing the number of medication errors at a pharmacy.
The Central Limit Theorem describes how the sampling distribution of sample means approaches a normal distribution as sample size increases, even if the population is not normally distributed. Specifically, it states that the sampling distribution of sample means will be approximately normally distributed whenever the sample size is large, and the larger the sample, the better the normal approximation. The Central Limit Theorem also predicts that the mean of the sampling distribution will equal the population mean, and the standard deviation of the sampling distribution will equal the population standard deviation divided by the square root of the sample size.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
A random variable is a variable whose values are determined by the outcome of a random experiment and can be used to model probabilities. Examples of random variables include the sum of dice rolls or number of heads from coin tosses. A probability distribution assigns probabilities to each possible value of a random variable. It must satisfy the properties that probabilities are greater than or equal to 0 and sum to 1. Common probability distributions include the binomial and normal distributions.
Chi‑square Test and its Application in Hypothesis Testing
Rakesh Rana, Richa Singhal
Statistical Section, Central Council for Research in Ayurvedic Sciences, Ministry of AYUSH, GOI, New Delhi, India
This document discusses probability distributions and binomial distributions. It defines:
i) Discrete and continuous probability distributions.
ii) The binomial distribution properties including the number of trials (n), probability of success (p), and probability of failure (q).
iii) How to calculate the mean, variance, and standard deviation of a binomial distribution using the formulas: mean=np, variance=npq, and standard deviation=√(npq).
This chapter discusses additional topics related to estimation, including forming confidence intervals for differences between dependent and independent population means and proportions. It provides formulas and examples for confidence intervals when population variances are known or unknown, and when variances are assumed equal or unequal. The chapter goals are to enable readers to compute confidence intervals and determine sample sizes for a variety of estimation problems involving means, proportions, differences between populations, and variances.
This chapter discusses hypothesis testing for differences between population means and variances. It covers tests for differences between two related population means using matched pairs, differences between two independent population means when variances are known and unknown, and tests of differences between two population variances. The key test statistics are t-tests and z-tests, and the chapter presents the assumptions, test statistics, and decision rules for each hypothesis test.
This chapter discusses hypothesis testing for differences between population means and variances. It covers testing the difference between two related population means using matched pairs. It also covers testing the difference between two independent population means when the population variances are known, unknown but assumed equal, and unknown but assumed unequal. Decision rules for lower-tail, upper-tail, and two-tail tests are provided for each case.
This chapter discusses methods for constructing confidence intervals for differences and comparisons between population parameters using sample data. It covers constructing confidence intervals for the difference between two independent population means when the standard deviations are known or unknown. It also addresses constructing confidence intervals when the population variances are assumed to be equal or unequal. The chapter concludes with constructing confidence intervals for the difference between two independent population proportions.
This chapter discusses additional sampling methods including stratified sampling, cluster sampling, and two-phase sampling. It provides formulas for estimating population means, totals, and proportions from stratified and cluster samples. Methods for determining optimal sample sizes to achieve desired levels of precision are also presented. Finally, the chapter addresses non-probability sampling techniques and their limitations compared to probability samples.
This chapter discusses methods for forming confidence intervals and conducting hypothesis tests to compare two population parameters, such as means, proportions, or variances. It covers topics like confidence intervals for the difference between two independent population means when the variances are known or unknown, confidence intervals for dependent sample means from before-after studies, and confidence intervals for comparing two independent population proportions. Examples are provided to demonstrate how to calculate confidence intervals for differences in means using pooled variances and how to form confidence intervals to compare proportions from two populations.
This document section discusses estimating population means from sample data. It presents methods for constructing confidence intervals for the population mean using the sample mean and standard deviation. Whether the t-distribution or normal distribution is used depends on whether the population standard deviation is known. Examples are provided to illustrate calculating margins of error and interpreting confidence intervals. The key requirements are that the sample be randomly selected and the population be normally distributed or the sample size be greater than 30.
The document provides an overview of how to conduct a t-test for independent means. It explains that this test is used to compare the means of two independent groups and determines if any difference observed could have been due to chance. It outlines the steps for this test, including calculating the pooled variance estimate, figuring the variance of each group's distribution of means, determining the variance and standard deviation of the distribution of differences between means, and computing the t-score to compare to critical values from the t-table. An example is also provided to demonstrate how to perform a t-test for independent means on sample data.
This document provides an overview of key concepts in statistics, including:
- Statistics helps deal with uncertainty and incomplete information in decision making.
- Descriptive statistics summarize and describe data, while inferential statistics make predictions from samples.
- There are different types of data (categorical, numerical/discrete, continuous) that influence analysis methods.
- Measures of central tendency like the mean, median, and mode describe typical values in a dataset.
- Measures of variability like the range, variance, and standard deviation describe how spread out values are.
Aron chpt 9 ed t test independent samplesKaren Price
This document describes the t-test for independent means, which is used to compare the means of two independent groups when population variances are unknown. It involves calculating the variance of each group, pooling the variances to estimate the population variance, and determining the variance and standard deviation of the distribution of differences between the two group means. The t-value is then calculated and compared to critical values from the t-distribution to determine if the group means are significantly different.
Aron chpt 9 ed t test independent samplesKaren Price
The document describes the t-test for independent means, which is used to compare the means of two independent groups when population variances are unknown. It involves calculating the variance of each group, pooling the variances to estimate the population variance, and determining the variance of the distribution of differences between the two sample means. The t-value is then computed and compared to critical values from the t-distribution to determine if the null hypothesis that the two population means are equal can be rejected.
Lecture 5 Sampling distribution of sample mean.pptxshakirRahman10
Objectives:
Distinguish between the distribution of population and distribution of its sample means
Explain the importance of central limit
theorem
Compute and interpret the standard error of the mean.
Sampling distribution of
sample mean:
A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes.
Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected.
For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population.
The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter. In the above example the average income of households would be the parameter.
The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean.
Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable.
Random variables have distribution ,and since the sample mean is a random variable it must have a distribution.
If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution.
Consider the population with population mean = μ
and standard deviation = σ.
Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means.
This distribution is called the Sampling Distribution of Means.
Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population.
Standard error of the
mean:
The quantity σ is referred to as the standard deviation .it is a measure of spread in the population .
The quality σ/n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean
A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem .
Central Limit Theorem:
Consider a population with finite mean and standard deviation . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples)
Simulation plays important role in many problems of our daily life. There has been increasing interest in the use of simulation to teach the concept of sampling distribution. In this paper we try to show the sampling distribution of some important statistic we often found in statistical methods by taking 10,000 simulations. The simulation is presented using R-programming language to help students to understand the concept of sampling distribution. This paper helps students to understand the concept of central limit theorem, law of large number and simulation of distribution of some important statistic we often encounter in statistical methods. This paper is about one sample and two sample inference. The paper shows the convergence of t-distribution to standard normal distribution. The sum of the square of deviations of items from population mean and sample mean follow chi-square distribution with different degrees of freedom. The ratio of two sample variance follow F-distribution. It is interesting that in linear regression the sampling distribution of the estimated parameters are normally distributed.
The ppt gives an idea about basic concept of Estimation. point and interval. Properties of good estimate is also covered. Confidence interval for single means, difference between two means, proportion and difference of two proportion for different sample sizes are included along with case studies.
This document discusses an analysis of variance (ANOVA) study conducted by Burke Marketing Services to evaluate potential new versions of a children's dry cereal. The experimental design and ANOVA were used to test differences between the cereal versions and make a product recommendation. The document provides an introduction to ANOVA, including how it can test for differences between three or more population means. It also outlines the assumptions of ANOVA, how to calculate test statistics like mean squares, and how to conduct an F-test to determine whether population means are equal or not.
Similar to Chap08 estimation additional topics (20)
Bab ini membahas dua pertanyaan tentang kebijakan stabilisasi makroekonomi: (1) apakah kebijakan sebaiknya aktif atau pasif, dan (2) apakah kebijakan sebaiknya dijalankan berdasarkan aturan atau kebijaksanaan. Pendukung kebijakan aktif berargumen bahwa fluktuasi ekonomi dapat dikurangi, sementara pendukung pasif lebih khawatir tentang ketidakefektifan dan ketid
Dokumen tersebut membahas tentang uang beredar dan permintaan uang. Secara singkat, dokumen tersebut menjelaskan bagaimana sistem perbankan "menciptakan" uang melalui pinjaman bank, tiga instrumen kebijakan moneter yang digunakan oleh The Fed untuk mengendalikan jumlah uang beredar, serta dua teori utama mengenai permintaan uang yaitu teori portofolio dan teori transaksi.
Bab ini membahas teori-teori utama konsumsi, termasuk hipotesis Keynes tentang pengaruh pendapatan saat ini terhadap konsumsi, model pilihan antarwaktu Irving Fisher, hipotesis siklus hidup Franco Modigliani, hipotesis pendapatan permanen Milton Friedman, dan implikasi teori-teori tersebut terhadap perilaku konsumsi.
Bab 15 membahas utang pemerintah, termasuk tingkat utang berbagai negara, pandangan tradisional dan Ricardian terhadap utang, dan perspektif lain seperti anggaran berimbang versus kebijakan fiskal optimal."
Ringkasan dari dokumen tersebut adalah:
1. Dokumen tersebut membahas model Mundell-Fleming dan rejim nilai tukar untuk perekonomian terbuka kecil.
2. Model Mundell-Fleming menggunakan kurva IS dan LM untuk menganalisis efek kebijakan fiskal, moneter, dan perdagangan di bawah sistem nilai tukar mengambang dan tetap.
3. Dokumen tersebut juga membahas penyebab perbedaan suku bunga antara d
Bab ini membahas bagaimana model Solow dapat diperluas untuk menggabungkan kemajuan teknologi, temuan empiris tentang pertumbuhan ekonomi, dan kebijakan untuk mendorong pertumbuhan. Topik utama termasuk bagaimana kemajuan teknologi dapat dimasukkan ke dalam model Solow, bukti konvergensi pendapatan antar negara, dan kebijakan untuk meningkatkan tingkat tabungan dan mengalokasikan investasi.
Dokumen tersebut membahas tentang perekonomian terbuka dan model perekonomian terbuka kecil, termasuk identitas akuntansi, faktor-faktor yang mempengaruhi neraca perdagangan dan nilai tukar, serta dampak kebijakan fiskal dan permintaan investasi terhadap variabel-variabel makroekonomi.
Bab ini membahas model pertumbuhan ekonomi Solow dan bagaimana tingkat tabungan dan pertumbuhan penduduk mempengaruhi standar hidup jangka panjang suatu negara. Model Solow menunjukkan bahwa negara dengan tingkat tabungan yang lebih tinggi akan memiliki tingkat modal dan pendapatan per kapita yang lebih tinggi dalam jangka panjang."
Dokumen tersebut membahas konsep-konsep data makroekonomi penting seperti Produk Domestik Bruto, indeks harga konsumen, dan tingkat pengangguran. Produk Domestik Bruto didefinisikan sebagai total pengeluaran untuk barang dan jasa yang diproduksi dalam negeri, sedangkan indeks harga konsumen digunakan untuk mengukur tingkat inflasi.
This document provides an overview of key statistical analysis techniques used in research methods, including descriptive statistics, validity testing, reliability testing, hypothesis testing, and techniques for comparing means such as t-tests and ANOVA. Descriptive statistics like mean and standard deviation are used to summarize variables measured on interval/ratio scales, while frequency and percentage summarize nominal/ordinal scales. Validity is assessed through exploratory factor analysis (EFA) to establish underlying dimensions. Reliability is measured using Cronbach's alpha. Hypothesis testing involves stating null and alternative hypotheses and making decisions based on statistical tests and p-values. T-tests compare two means and ANOVA compares three or more means, both assuming equal variances based on Levene
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...