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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.

Ask by Stuart Cook. in the United States
Mar 14,2025

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Answer

The probability is approximately 28.77%.

Solution

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 \( \boxed{28.77\%} \).

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To find the probability of observing \( 20\% \) or less users aged \( 30-39 \) among a sample of \( 185 \) TikTok users where the proportion is \( 21.7\% \): 1. **Calculate the mean (\( p \))**: \[ p = 0.217 \] Thus, the mean number of users aged \( 30-39 \) in the sample: \[ \mu = n \cdot p = 185 \cdot 0.217 = 40.195 \] 2. **Calculate the standard deviation (\( \sigma \))**: \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{185 \cdot 0.217 \cdot (1 - 0.217)} \approx \sqrt{185 \cdot 0.217 \cdot 0.783} \approx \sqrt{25.1886} \approx 5.0187 \] 3. **Find the critical value for \( 20\% \)**: The number of users that represents \( 20\% \) of the sample: \[ X = 0.20 \cdot 185 = 37 \] 4. **Convert to a Z-score**: \[ Z = \frac{X - \mu}{\sigma} = \frac{37 - 40.195}{5.0187} \approx \frac{-3.195}{5.0187} \approx -0.6367 \] 5. **Find the probability**: Using the Z-table or normal distribution calculator for \( Z = -0.6367 \), the cumulative probability \( P(Z \leq -0.6367) \approx 0.2632 \). Thus, the probability of observing \( 20\% \) or less users aged \( 30-39 \) is approximately \( 26.32\% \).

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