5.4 Homework
Question 9, 5.4.33
HW Score: $59.62 \%, 7.75$ of 13 points
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The height of women ages $20-29$ is normally distributed, with a mean of 64.2 inches. Assume $\sigma=2.7$ inches. Are you more likely to randomly select 1 woman with a height less than 66.4 inches or are you more likely to select a sample of 14 women with a mean height less than 66.4 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 66.4 inches?
(Round to four decimal places as needed.)

Step 1 ：The problem is asking for the probability of selecting a woman with a height less than 66.4 inches. This is a problem of normal distribution.

Step 2 ：We can calculate the z-score for the height 66.4 inches and then find the corresponding probability from the standard normal distribution table.

Step 3 ：The z-score is calculated as \( (X - \mu) / \sigma \), where X is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 4 ：In this case, \(X = 66.4\), \(\mu = 64.2\), and \(\sigma = 2.7\).

Step 5 ：Calculate the z-score: \( z = (X - \mu) / \sigma = (66.4 - 64.2) / 2.7 = 0.8148 \).

Step 6 ：Use the cumulative distribution function to find the probability corresponding to the z-score.

Step 7 ：The probability of randomly selecting 1 woman with a height less than 66.4 inches is approximately 0.7924.

Step 8 ：Final Answer: The probability of randomly selecting 1 woman with a height less than 66.4 inches is \(\boxed{0.7924}\).