Answered: Assume that a sample is used to…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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$**Title: Calculating Confidence Interval for Population Proportion** **Introduction:** When you use a sample to estimate the population proportion $ p $ (for example, the proportion of people who prefer a certain brand), it's essential to determine how accurate this estimate is. This accuracy is often expressed as a confidence interval. Here, you'll learn how to find the 99% confidence interval for the population proportion using a given sample. **Example Problem:** Assume that a sample is used to estimate a population proportion $ p $. We need to find the 99% confidence interval for a sample of size 231 with 116 successes. Your answer should be in the form of a tri-linear inequality using decimals, not percentages, and should be accurate to three decimal places. **Steps:** 1. **Determine the sample proportion, $\hat{p}$:** \[ \hat{p} = \frac{x}{n} \] Where $ x $ is the number of successes and $ n $ is the sample size. \[ \hat{p} = \frac{116}{231} \] 2. **Calculate the standard error (SE) for the proportion:** \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] 3. **Determine the Z-value for a 99% confidence level:** The Z-value (critical value) for a 99% confidence interval is approximately 2.576. This value is derived from standard normal distribution tables. 4. **Calculate the margin of error (ME):** \[ ME = Z \times SE \] 5. **Determine the confidence interval:** \[ CI = \hat{p} \pm ME \] 6. **Express your confidence interval as a tri-linear inequality:** \[ \text{Lower Bound} < p < \text{Upper Bound} \] **Interactive Exercise:** Below are provided spaces to calculate and input your answer for the confidence interval: \[ \underline{\hspace{2cm}} < p < \underline{\hspace{2cm}} \] By following these steps, you will determine the population proportion $ p $ with 99% confidence. **Conclusion:** Understanding confidence intervals is a vital concept in statistics that allows you to estimate$

Transcribed Image Text:**Title: Calculating Confidence Interval for Population Proportion** **Introduction:** When you use a sample to estimate the population proportion $ p $ (for example, the proportion of people who prefer a certain brand), it's essential to determine how accurate this estimate is. This accuracy is often expressed as a confidence interval. Here, you'll learn how to find the 99% confidence interval for the population proportion using a given sample. **Example Problem:** Assume that a sample is used to estimate a population proportion $ p $. We need to find the 99% confidence interval for a sample of size 231 with 116 successes. Your answer should be in the form of a tri-linear inequality using decimals, not percentages, and should be accurate to three decimal places. **Steps:** 1. **Determine the sample proportion, $\hat{p}$:** \[ \hat{p} = \frac{x}{n} \] Where $ x $ is the number of successes and $ n $ is the sample size. \[ \hat{p} = \frac{116}{231} \] 2. **Calculate the standard error (SE) for the proportion:** \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] 3. **Determine the Z-value for a 99% confidence level:** The Z-value (critical value) for a 99% confidence interval is approximately 2.576. This value is derived from standard normal distribution tables. 4. **Calculate the margin of error (ME):** \[ ME = Z \times SE \] 5. **Determine the confidence interval:** \[ CI = \hat{p} \pm ME \] 6. **Express your confidence interval as a tri-linear inequality:** \[ \text{Lower Bound} < p < \text{Upper Bound} \] **Interactive Exercise:** Below are provided spaces to calculate and input your answer for the confidence interval: \[ \underline{\hspace{2cm}} < p < \underline{\hspace{2cm}} \] By following these steps, you will determine the population proportion $ p $ with 99% confidence. **Conclusion:** Understanding confidence intervals is a vital concept in statistics that allows you to estimate

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