High blood pressure has been identified as a risk factor for heart attacks and strokes. The proportion of U.S. adults with high blood pressure is 0.4 . A sample of 40 U.S. adults is chosen. Use the TI-84 Plus calculator as needed. Round the answers to at least four decimal places as needed. Find the probability that more than 42% of the people in this sample have high blood pressure.

We are dealing with a **binomial distribution** problem here because we are sampling a fixed number of individuals (40 U.S. adults), and each individual either has high blood pressure or not. However, when the sample size is large, as in this case, the **normal approximation** to the binomial distribution is often used.

### Step 1: Define the parameters

- The proportion (probability of success, \( p \)) of U.S. adults with high blood pressure is given as \( p = 0.4 \).
- The sample size is \( n = 40 \).
- We are asked to find the probability that more than 42% of the sample has high blood pressure, meaning we are looking for \( P(\hat{p} > 0.42) \), where \( \hat{p} \) is the sample proportion.

### Step 2: Compute the mean and standard deviation of the sample proportion

For a binomial distribution, the mean \( \mu_{\hat{p}} \) and standard deviation \( \sigma_{\hat{p}} \) of the sample proportion \( \hat{p} \) are given by:

\[
\mu_{\hat{p}} = p = 0.4
\]
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.4(1 - 0.4)}{40}} = \sqrt{\frac{0.24}{40}} = \sqrt{0.006} \approx 0.0775
\]

### Step 3: Convert the sample proportion to a z-score

We want to find the probability that \( \hat{p} > 0.42 \). To do this, we convert \( \hat{p} = 0.42 \) to a z-score using the formula:

\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}
\]

Substitute the values:

\[
z = \frac{0.42 - 0.4}{0.0775} \approx \frac{0.02}{0.0775} \approx 0.2581
\]

### Step 4: Use the z-table or

High blood pressure has been identified as a risk factor for heart attacks and strokes. The proportion of U.S. adults with high blood pressure is 0.4. A sample of 40 U.S. adults is chosen. Use the TI-84 Plus calculator as needed. Round the answers to at least four decimal places as needed. Find the probability that more than 42% of the people in this sample have high blood pressure.

Learn how to find the probability that more than 42% of a sample of U.S. adults have high blood pressure using normal approximation to the binomial distribution. Includes key concepts like Z-scores and normal distribution. Step-by-step solution for college-level statistics.

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