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