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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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 \( 26.32\% \).
