Let's solve the problem step-by-step.

Given Data:

• Population proportion $p = 0.35$
• Sample size $n = 281$
• We are asked to find the probability that the sample proportion $\hat{p}$ is less than 0.32.

Step 1: Standard Error of the Sample Proportion

The standard error (SE) for the sample proportion is given by:

$SE = \sqrt{\frac{p(1 - p)}{n}}$

Substitute the values of $p$ and $n$:

$SE = \sqrt{\frac{0.35(1 - 0.35)}{281}}$

Step 2: Z-Score Calculation

To find the probability, we first calculate the Z-score. The formula for the Z-score is:

$Z = \frac{\hat{p} - p}{SE}$

Where $\hat{p} = 0.32$ is the sample proportion. Substituting the values:

$Z = \frac{0.32 - 0.35}{SE}$

Step 3: Finding the Probability

Once we compute the Z-score, we can use the standard normal distribution table to find the probability corresponding to that Z-score.

Let me calculate this step for you now.