In a random survey of 406 people, 295 indicated that they prefered to go a forest to camp. Find a 83% confidence interval for the true proportion of all people who prefer to go to a forest to camp. (Round answers to three decimal places.) Error =
In a random survey of 406 people, 295 indicated that they prefered to go a forest to camp. Find a 83% confidence interval for the true proportion of all people who prefer to go to a forest to camp. (Round answers to three decimal places.) Error =
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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![**Calculating Confidence Intervals for Proportions**
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In a random survey of 406 people, 295 indicated that they preferred to go to a forest to camp. Find an 83% confidence interval for the true proportion of all people who prefer to go to a forest to camp. (Round answers to three decimal places.)
\[ \text{Error} = \_\_\_\_\_\_ \]
\[ \_\_\_\_\_ < p < \_\_\_\_\_ \]
### Steps to Calculate the Confidence Interval
To find the confidence interval for the true proportion of the population that prefers to go to a forest to camp, follow these steps:
1. **Calculate the sample proportion (p̂):**
\[ \hat{p} = \frac{\text{Number of favorable outcomes}}{\text{Total number surveyed}} = \frac{295}{406} \]
2. **Find the standard error (SE):**
\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Where:
- \(\hat{p}\) is the sample proportion
- \(n\) is the sample size
3. **Determine the Z-value associated with the desired confidence level.**
For an 83% confidence level, look up the Z value in standard normal distribution tables.
4. **Calculate the margin of error (ME):**
\[ ME = Z \times SE \]
5. **Determine the lower and upper bounds of the confidence interval:**
\[ \hat{p} - ME < p < \hat{p} + ME \]
By plugging in the appropriate values, you can determine the exact error margin and confidence interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff21c709c-710e-425a-a9e1-f3b584be2fe0%2Fffb1bea3-3eb6-4053-b553-d26c1f7167c9%2F2jm91l_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculating Confidence Intervals for Proportions**
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In a random survey of 406 people, 295 indicated that they preferred to go to a forest to camp. Find an 83% confidence interval for the true proportion of all people who prefer to go to a forest to camp. (Round answers to three decimal places.)
\[ \text{Error} = \_\_\_\_\_\_ \]
\[ \_\_\_\_\_ < p < \_\_\_\_\_ \]
### Steps to Calculate the Confidence Interval
To find the confidence interval for the true proportion of the population that prefers to go to a forest to camp, follow these steps:
1. **Calculate the sample proportion (p̂):**
\[ \hat{p} = \frac{\text{Number of favorable outcomes}}{\text{Total number surveyed}} = \frac{295}{406} \]
2. **Find the standard error (SE):**
\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Where:
- \(\hat{p}\) is the sample proportion
- \(n\) is the sample size
3. **Determine the Z-value associated with the desired confidence level.**
For an 83% confidence level, look up the Z value in standard normal distribution tables.
4. **Calculate the margin of error (ME):**
\[ ME = Z \times SE \]
5. **Determine the lower and upper bounds of the confidence interval:**
\[ \hat{p} - ME < p < \hat{p} + ME \]
By plugging in the appropriate values, you can determine the exact error margin and confidence interval.
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