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.

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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 $$ \boxed{28.77\%} $$.
The probability is approximately 28.77%.

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