The proportion of TikTok users ages \( 30-39 \) is estimated to be \( 21.7 \% \). In a randomly selected sample of 185 TikTok users, find the probability that you observe \( 20 \% \) or less users aged \( 30-39 \). You may use the Normal Distribution tool. Mean should not be rounded, standard deviation may be rounded to four decimal places. Rercentage, include the \% symbol.

Let \( p = 0.217 \) be the proportion of TikTok users ages \( 30-39 \) and \( n = 185 \) be the sample size. We want the probability that the observed proportion \( \hat{p} \) is \( 20\% \) or less, i.e., \( \hat{p} \leq 0.20 \). **Step 1. Compute the mean and standard deviation of \(\hat{p}\):** The mean of the sampling distribution of \(\hat{p}\) is: \[ \mu_{\hat{p}} = p = 0.217 \] The standard deviation (or standard error) is: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.217 \times (1-0.217)}{185}} \] Calculate the numerator: \[ 0.217 \times 0.783 = 0.169731 \] Thus, \[ \sigma_{\hat{p}} = \sqrt{\frac{0.169731}{185}} \approx \sqrt{0.000917} \approx 0.0303 \] **Step 2. Standardize the value \( \hat{p} = 0.20 \):** The \( z \)-score is given by: \[ z = \frac{0.20 - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.20 - 0.217}{0.0303} \approx \frac{-0.017}{0.0303} \approx -0.5611 \] **Step 3. Find the probability corresponding to \( z = -0.5611 \):** Using the standard normal distribution, \[ P(Z \leq -0.5611) \approx 0.2877 \] **Step 4. Express the probability as a percentage:** \[ 0.2877 \times 100\% \approx 28.77\% \] Thus, the probability that \( 20\% \) or less TikTok users in the sample are aged \( 30-39 \) is approximately \( \boxed{28.77\%} \).