the given degree of confidence and sample data to construct a confidence interval for the population proportion p. Round to three decimal digits needed. Of 147 randomly selected adults, 34 were found to have high blood pressure. Construct a 95% confidence interval for the true percentage of all Its that have high blood pressure.
the given degree of confidence and sample data to construct a confidence interval for the population proportion p. Round to three decimal digits needed. Of 147 randomly selected adults, 34 were found to have high blood pressure. Construct a 95% confidence interval for the true percentage of all Its that have high blood pressure.
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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![### Constructing a 95% Confidence Interval for Population Proportion
In this example, we are given the following data:
- A sample of 147 randomly selected adults.
- Out of these, 34 were found to have high blood pressure.
The task is to construct a 95% confidence interval for the true percentage of all adults that have high blood pressure. The confidence interval will be rounded to three decimal places as required.
#### Step-by-Step Solution:
1. **Sample Proportion (p̂):**
\[ \hat{p} = \frac{x}{n} = \frac{34}{147} \]
Calculating this gives:
\[ \hat{p} \approx 0.231 (rounded to three decimal places) \]
2. **Standard Error (SE):**
\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Plugging in the numbers:
\[ SE = \sqrt{\frac{0.231 \times (1 - 0.231)}{147}} \approx 0.035 (rounded to three decimal places) \]
3. **Critical Value (Z*) for a 95% Confidence Interval:**
The critical value for a 95% confidence level is approximately 1.96.
4. **Confidence Interval Calculation:**
\[ \text{Margin of Error (ME)} = Z^* \times SE = 1.96 \times 0.035 \approx 0.069 \]
Therefore, the 95% confidence interval is:
\[ \hat{p} \pm \text{ME} \]
\[ 0.231 \pm 0.069 \]
5. **Result:**
\[ \text{Confidence Interval} = (0.162, 0.300) \]
So, the 95% confidence interval for the true percentage of all adults that have high blood pressure is approximately (0.162, 0.300). This means we are 95% confident that the true proportion of adults with high blood pressure lies between 16.2% and 30.0%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78a9a2c8-87fa-4dce-a05a-3dd826ae3dea%2Fc8633f98-c3a8-4bee-8583-8e9205212cb7%2F3vrjbp_processed.png&w=3840&q=75)
Transcribed Image Text:### Constructing a 95% Confidence Interval for Population Proportion
In this example, we are given the following data:
- A sample of 147 randomly selected adults.
- Out of these, 34 were found to have high blood pressure.
The task is to construct a 95% confidence interval for the true percentage of all adults that have high blood pressure. The confidence interval will be rounded to three decimal places as required.
#### Step-by-Step Solution:
1. **Sample Proportion (p̂):**
\[ \hat{p} = \frac{x}{n} = \frac{34}{147} \]
Calculating this gives:
\[ \hat{p} \approx 0.231 (rounded to three decimal places) \]
2. **Standard Error (SE):**
\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Plugging in the numbers:
\[ SE = \sqrt{\frac{0.231 \times (1 - 0.231)}{147}} \approx 0.035 (rounded to three decimal places) \]
3. **Critical Value (Z*) for a 95% Confidence Interval:**
The critical value for a 95% confidence level is approximately 1.96.
4. **Confidence Interval Calculation:**
\[ \text{Margin of Error (ME)} = Z^* \times SE = 1.96 \times 0.035 \approx 0.069 \]
Therefore, the 95% confidence interval is:
\[ \hat{p} \pm \text{ME} \]
\[ 0.231 \pm 0.069 \]
5. **Result:**
\[ \text{Confidence Interval} = (0.162, 0.300) \]
So, the 95% confidence interval for the true percentage of all adults that have high blood pressure is approximately (0.162, 0.300). This means we are 95% confident that the true proportion of adults with high blood pressure lies between 16.2% and 30.0%.
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