 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 0.3. A sample of 32 U.S. adults is chosen. Use the TI-84 Plus Calculator as needed. Round the answer to at least four decimal places. Find the probability that less than 23% of the people in the sample of 78 have high blood pressure.
The probability that less than 23% of the people in the sample of 78 have high blood pressure is

To solve this problem, we need to use the normal approximation to the binomial distribution since the sample size is relatively large. Here are the steps:

### Step 1: Define the problem
- Proportion of U.S. adults with high blood pressure: \( p = 0.30 \)
- Sample size: \( n = 78 \)
- We are asked to find the probability that less than 23% of the sample has high blood pressure. So, we need to calculate the probability for \( \hat{p} < 0.23 \), where \( \hat{p} \) is the sample proportion.

### Step 2: Calculate the mean and standard deviation
Using the formulas for the mean and standard deviation of the sample proportion:

\[
\mu_{\hat{p}} = p = 0.30
\]
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.30(1 - 0.30)}{78}} = \sqrt{\frac{0.30 \times 0.70}{78}} \approx 0.0512
\]

### Step 3: Standardize the value
We need to find the z-score for \( \hat{p} = 0.23 \):

\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.23 - 0.30}{0.0512} = \frac{-0.07}{0.0512} \approx -1.3672
\]

### Step 4: Find the probability
Using the z-score of \( -1.3672 \), we now look up the cumulative probability in the z-table (or use a calculator). The cumulative probability for a z-score of \( -1.3672 \) is approximately 0.0855.

### Final Answer:
The probability that less than 23% of the people in the sample have high blood pressure is approximately **0.0855**.

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Would you like further details or have any questions on this? Here are some related questions to expand your understanding:

1. What is the binomial distribution and when is it approximated by the normal distribution?
2. How do you calculate the mean and standard deviation of a sample proportion?
3. What is the significance of the z-score in statistics?
4. How do you interpret probabilities in a real-world context such as this one?
5. How does increasing the sample size affect the accuracy of the normal approximation?

**Tip:** The larger the sample size, the more the binomial distribution resembles a normal distribution, which makes the normal approximation more accurate.

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 0.3. A sample of 32 U.S. adults is chosen. Use the TI-84 Plus Calculator as needed. Round the answer to at least four decimal places. Find the probability that less than 23% of the people in the sample of 78 have high blood pressure.

Learn how to calculate the probability that less than 23% of a sample of 78 U.S. adults have high blood pressure. This problem involves key statistical concepts, including the normal approximation to the binomial distribution, z-scores, and sample proportions. Suitable for students in Grades 10-12.

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