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.
When you perform a hypothesis test in statistics, a p-value helps you determine the significance of your results. ... The p-value is a number between 0 and 1 and interpreted in the following way: A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.
When you perform a hypothesis test in statistics, a p-value helps you determine the significance of your results. ... The p-value is a number between 0 and 1 and interpreted in the following way: A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.
Explains use of statistical power, inferential decision making, effect sizes, confidence intervals in applied social science research, and addresses the issue of publication bias and academic integrity.
Confidence Interval ModuleOne of the key concepts of statist.docxmaxinesmith73660
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval.
The formula that we use to calculate the standard error of the mean is:
s.e. = s / √N – 1
where s = the standard deviation calculated from the sample; and
N = the sample size.
So the formula tells us that the standard error of the mean is equal to the
standard deviation divided by the square root of the sample size minus 1.
This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected.
THE CONFIDENCE INTERVAL (CI)
The formula for the CI is a function of the sample size (N).
For samples sizes ≥ 100, the formula for the CI is:
CI = (the sample mean) + & - Z(s.e.).
Let’s look at an example to see how this formula works.
* Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link.
Example 1
Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.
1. Illustrate point and interval estimations.
2. Distinguish between point and interval estimation.
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Standard Error & Confidence Intervals.pptxhanyiasimple
Certainly! Let's delve into the concept of **standard error**.
## What Is Standard Error?
The **standard error (SE)** is a statistical measure that quantifies the **variability** between a sample statistic (such as the mean) and the corresponding population parameter. Specifically, it estimates how much the sample mean would **vary** if we were to repeat the study using **new samples** from the same population. Here are the key points:
1. **Purpose**: Standard error helps us understand how well our **sample data** represents the entire population. Even with **probability sampling**, where elements are randomly selected, some **sampling error** remains. Calculating the standard error allows us to estimate the representativeness of our sample and draw valid conclusions.
2. **High vs. Low Standard Error**:
- **High Standard Error**: Indicates that sample means are **widely spread** around the population mean. In other words, the sample may not closely represent the population.
- **Low Standard Error**: Suggests that sample means are **closely distributed** around the population mean, indicating that the sample is representative of the population.
3. **Decreasing Standard Error**:
- To decrease the standard error, **increase the sample size**. Using a large, random sample minimizes **sampling bias** and provides a more accurate estimate of the population parameter.
## Standard Error vs. Standard Deviation
- **Standard Deviation (SD)**: Describes variability **within a single sample**. It can be calculated directly from sample data.
- **Standard Error (SE)**: Estimates variability across **multiple samples** from the same population. It is an **inferential statistic** that can only be estimated (unless the true population parameter is known).
### Example:
Suppose we have a random sample of 200 students, and we calculate the mean math SAT score to be 550. In this case:
- **Sample**: The 200 students
- **Population**: All test takers in the region
The standard error helps us understand how well this sample represents the entire population's math SAT scores.
Remember, the standard error is crucial for making valid statistical inferences. By understanding it, researchers can confidently draw conclusions based on sample data. 📊🔍
If you need further clarification or have additional questions, feel free to ask! 😊
---
I've provided a concise explanation of standard error, emphasizing its importance in statistical analysis. If you'd like more details or specific examples, feel free to ask! ¹²³⁴
Source: Conversation with Copilot, 5/31/2024
(1) What Is Standard Error? | How to Calculate (Guide with Examples) - Scribbr. https://www.scribbr.com/statistics/standard-error/.
(2) Standard Error (SE) Definition: Standard Deviation in ... - Investopedia. https://www.investopedia.com/terms/s/standard-error.asp.
(3) Standard error Definition & Meaning - Merriam-Webster. https://www.merriam-webster.com/dictionary/standard%20error.
(4) Standard err
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
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2. § 6.1 Confidence Intervals for the Mean (Large Samples)
3. Point Estimate for Population μ A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean, , is the sample mean, Example : A random sample of 32 textbook prices (rounded to the nearest dollar) is taken from a local college bookstore. Find a point estimate for the population mean, . The point estimate for the population mean of textbooks in the bookstore is $74.22. 101 88 67 45 110 90 68 45 121 98 98 96 95 94 90 87 87 87 86 79 74 67 66 65 65 56 54 45 45 38 34 34
4. Interval Estimate An interval estimate is an interval, or range of values, used to estimate a population parameter. How confident do we want to be that the interval estimate contains the population mean, μ ? Point estimate for textbooks • 74.22 interval estimate
5. Level of Confidence The level of confidence c is the probability that the interval estimate contains the population parameter. c is the area beneath the normal curve between the critical values. The remaining area in the tails is 1 – c . Use the Standard Normal Table to find the corresponding z - scores. z z = 0 z c z c Critical values c (1 – c ) (1 – c )
6. Common Levels of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean, μ . The corresponding z - scores are ± 1.645. z c = 1.645 z c = 1.645 z z = 0 z c z c 0.90 0.05 0.05
7. Common Levels of Confidence If the level of confidence is 95%, this means that we are 95% confident that the interval contains the population mean, μ . The corresponding z - scores are ± 1.96. z c = 1.96 z c = 1.96 z z = 0 z c z c 0.95 0.025 0.025
8. Common Levels of Confidence If the level of confidence is 99%, this means that we are 99% confident that the interval contains the population mean, μ . The corresponding z - scores are ± 2.575. z c = 2.575 z c = 2.575 z z = 0 z c z c 0.99 0.005 0.005
9. Margin of Error The difference between the point estimate and the actual population parameter value is called the sampling error . Given a level of confidence, the margin of error (sometimes called the maximum error of estimate or error tolerance) E is the greatest possible distance between the point estimate and the value of the parameter it is estimating. When n 30, the sample standard deviation, s , can be used for . When μ is estimated, the sampling error is the difference μ – . Since μ is usually unknown, the maximum value for the error can be calculated using the level of confidence.
10. Margin of Error Example : A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is = 74.22, and the sample standard deviation is s = 23.44. Use a 95% confidence level and find the margin of error for the mean price of all textbooks in the bookstore. We are 95% confident that the margin of error for the population mean (all the textbooks in the bookstore) is about $8.12. Since n 30, s can be substituted for σ.
11. Confidence Intervals for μ A c - confidence interval for the population mean μ is The probability that the confidence interval contains μ is c . Continued. Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. Example : A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is = 74.22, the sample standard deviation is s = 23.44, and the margin of error is E = 8.12.
12. Confidence Intervals for μ Example continued : Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. s = 23.44 E = 8.12 With 95% confidence we can say that the cost for all textbooks in the bookstore is between $66.10 and $82.34. = 66.1 = 82.34 Left endpoint = ? Right endpoint = ? • • •
13.
14. Confidence Intervals for μ ( Known) Example : A random sample of 25 students had a grade point average with a mean of 2.86. Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Construct a 90% confidence interval for the population mean grade point average. n = 25 = 0.15 z c = 1.645 2.81 < μ < 2.91 With 90% confidence we can say that the mean grade point average for all students in the population is between 2.81 and 2.91.
15. Sample Size Given a c - confidence level and a maximum error of estimate, E , the minimum sample size n , needed to estimate , the population mean, is If is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members. Example : You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? Continued.
16. Sample Size Example continued : You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? s = 23.44 z c = 2.575 You should include at least 146 books in your sample. (Always round up .)
20. Critical Values of t Example : Find the critical value t c for a 95% confidence when the sample size is 5. Appendix B: Table 5: t -Distribution Continued. d.f. = n – 1 = 5 – 1 = 4 c = 0.95 t c = 2.776 3.365 2.571 2.015 1.476 .727 5 3.747 2.776 2.132 1.533 .741 4 4.541 3.182 2.353 1.638 .765 3 6.965 4.303 2.920 1.886 .816 2 31.821 12.706 6.314 3.078 1.000 1 0.02 0.05 0.10 0.20 0.50 Two tails, d.f. 0.01 0.025 0.05 0.10 0.25 One tail, 0.98 0.95 0.90 0.80 0.50 Level of confidence, c
21. Critical Values of t Example continued : Find the critical value t c for a 95% confidence when the sample size is 5. 95% of the area under the t -distribution curve with 4 degrees of freedom lies between t = ±2.776. t t c = 2.776 t c = 2.776 c = 0.95
22.
23. Constructing a Confidence Interval Example : In a random sample of 20 customers at a local fast food restaurant, the mean waiting time to order is 95 seconds, and the standard deviation is 21 seconds. Assume the wait times are normally distributed and construct a 90% confidence interval for the mean wait time of all customers. We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds. s = 21 t c = 1.729 n = 20 d.f. = 19 86.9 < μ < 103.1
24. Normal or t -Distribution? Is n 30? Is the population normally, or approximately normally, distributed? You cannot use the normal distribution or the t -distribution. Is known? No Yes No Use the normal distribution with If is unknown, use s instead. Yes No Use the normal distribution with Yes Use the t - distribution with and n – 1 degrees of freedom.
25. Normal or t -Distribution? Example : Determine whether to use the normal distribution, the t - distribution, or neither. a.) n = 50, the distribution is skewed, s = 2.5 The normal distribution would be used because the sample size is 50. b.) n = 25, the distribution is skewed, s = 52.9 Neither distribution would be used because n < 30 and the distribution is skewed. c.) n = 25, the distribution is normal, = 4.12 The normal distribution would be used because although n < 30, the population standard deviation is known.
26. § 6.3 Confidence Intervals for Population Proportions
27. Point Estimate for Population p The probability of success in a single trial of a binomial experiment is p . This probability is a population proportion . The point estimate for p , the population proportion of successes, is given by the proportion of successes in a sample and is denoted by where x is the number of successes in the sample and n is the number in the sample. The point estimate for the proportion of failures is = 1 – The symbols and are read as “ p hat” and “ q hat.”
28. Point Estimate for Population p Example : In a survey of 1250 US adults, 450 of them said that their favorite sport to watch is baseball. Find a point estimate for the population proportion of US adults who say their favorite sport to watch is baseball. The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. n = 1250 x = 450
29. Confidence Intervals for p A c - confidence interval for the population proportion p is where The probability that the confidence interval contains p is c . Example : Construct a 90% confidence interval for the proportion of US adults who say baseball is their favorite sport to watch. Continued. n = 1250 x = 450
30. Confidence Intervals for p Example continued : With 90% confidence we can say that the proportion of all US adults who say baseball is their favorite sport to watch is between 33.8% and 38.2%. n = 1250 x = 450 Left endpoint = ? Right endpoint = ? • • •
31.
32. Sample Size Given a c - confidence level and a margin of error, E , the minimum sample size n , needed to estimate p is This formula assumes you have an estimate for and If not, use and Example : You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. Continued.
33. Sample Size You should sample at least 2213 adults to be 95% confident. (Always round up .) Example continued : You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. n = 1250 x = 450
34. § 6.4 Confidence Intervals for Variance and Standard Deviation
35. The Chi - Square Distribution The point estimate for 2 is s 2 , and the point estimate for is s . s 2 is the most unbiased estimate for 2 . You can use the chi - square distribution to construct a confidence interval for the variance and standard deviation. If the random variable x has a normal distribution, then the distribution of forms a chi - square distribution for samples of any size n > 1.
36.
37. Critical Values for X 2 There are two critical values for each level of confidence. The value χ 2 R represents the right - tail critical value and χ 2 L represents the left - tail critical value. The area between the left and right critical values is c . X 2 X 2 R Area to the right of X 2 R X 2 X 2 L Area to the right of X 2 L X 2 X 2 R X 2 L c
38. Critical Values for X 2 Example : Find the critical values χ 2 R and χ 2 L for an 80% confidence when the sample size is 18. Continued. Because the sample size is 18, there are d.f. = n – 1 = 18 – 1 = 17 degrees of freedom, Use the Chi - square distribution table to find the critical values. Area to the right of χ 2 R = Area to the right of χ 2 L =
40. Confidence Intervals for 2 and A c - confidence interval for a population variance and standard deviation is as follows. The probability that the confidence intervals contain 2 or is c . Confidence Interval for 2 : Confidence Interval for :
41.
42.
43. Constructing a Confidence Interval Example : You randomly select and weigh 41 samples of 16-ounce bags of potato chips. The sample standard deviation is 0.05 ounce. Assuming the weights are normally distributed, construct a 90% confidence interval for the population standard deviation. d.f. = n – 1 = 41 – 1 = 40 degrees of freedom, The critical values are χ 2 R = 55.758 and χ 2 L = 26.509. Continued. Area to the right of χ 2 R = Area to the right of χ 2 L =
44. Constructing a Confidence Interval Example continued : χ 2 R = 55.758 χ 2 L = 26.509 With 90% confidence we can say that the population standard deviation is between 0.04 and 0.06 ounces. Left endpoint = ? Right endpoint = ? • •