entraintint ineorem Part 2 of 3 score: \( 56.25 \%, 4.5 \) of 8 points Points: 0 of 1 Save The height of women ages \( 20-29 \) is normally distributed, with a mean of 64.3 inches. Assume \( \sigma=2.7 \) inches. Are you more likely to randomly select 1 wornan with a height less than 65.8 inches or are you more likely to select a sample of 21 women with a mean height less than 65.8 inches? Explain. Click the icon to view page 1 of the standard normal table. Click the icon to view page 2 of the standard normal table. What is the probability of randomly selecting 1 woman with a height less than 65.8 inches? 0.7123 (Round to four decimal places as needed.) What is the probability of selecting a sample of 21 women with a mean height less than 65.8 inches? (Round to four decimal places as needed.)

To determine whether you are more likely to select either a single woman or a sample of 21 women with a mean height less than 65.8 inches, you'll calculate the probability for both scenarios. For a single woman, you've already calculated that the probability \( P(X < 65.8) \) is approximately \( 0.7123 \). Now, for the sample of 21 women, you need to use the Central Limit Theorem, which states that the sampling distribution of the sample mean will also be normally distributed. The mean remains \( \mu = 64.3 \), but the standard deviation (standard error) must be adjusted for the sample size. The standard error \( \sigma_{\bar{x}} \) is calculated as: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{2.7}{\sqrt{21}} \approx 0.5885. \] Next, you will standardize the sample mean using the z-score formula: \[ z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{65.8 - 64.3}{0.5885} \approx 2.54. \] Now, you can look up \( z = 2.54 \) in the standard normal table, which gives you a probability \( P(Z < 2.54) \). This is approximately \( 0.9945 \). So, you're more likely to select a sample of 21 women with a mean height less than 65.8 inches, with a probability of \( 0.9945 \). 1. Probability of selecting a single woman with height less than 65.8 inches: \( 0.7123 \) 2. Probability of selecting a sample of 21 women with a mean height less than 65.8 inches: \( 0.9945 \)