Question

the height of women ages 20 - 29 is normally distributed, with a mean of 64.6 inches. assume \\( \sigma = 2.7 \\) inches. are you more likely to randomly select 1 woman with a height less than 67.1 inches or are you more likely to select a sample of 12 women with a mean height less than 67.1 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 67.1 inches? (round to four decimal places as needed.) what is the probability of selecting a sample of 12 women with a mean height less than 67.1 inches? (round to four decimal places as needed.) are you more likely to randomly select 1 woman with a height less than 67.1 inches or are you more likely to select a sample of 12 women with a mean height less than 67.1 inches? choose the correct answer below. a. it is more likely to select a sample of 12 women with a mean height less than 67.1 inches because the sample of 12 has a lower probability.

Explanation:

Step1: Calculate \( z \)-score for one woman

The formula for \( z \)-score is \( z=\frac{x-\mu}{\sigma} \). Here, \( x = 67.1 \), \( \mu=64.6 \), \( \sigma = 2.7 \).

$$ z=\frac{67.1 - 64.6}{2.7}=\frac{2.5}{2.7}\approx0.93 $$

Using the standard - normal table, \( P(X < 67.1)=P(Z<0.93) \). Looking up \( z = 0.93 \) in the standard - normal table, \( P(Z < 0.93)=0.8238 \).

Step2: Calculate \( z \)-score for the sample mean

The formula for the \( z \)-score of the sample mean is \( z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}} \). Here, \( \bar{x}=67.1 \), \( \mu = 64.6 \), \( \sigma=2.7 \), \( n = 12 \).

$$ z=\frac{67.1-64.6}{\frac{2.7}{\sqrt{12}}}=\frac{2.5}{\frac{2.7}{3.464}}\approx\frac{2.5}{0.78}\approx3.21 $$

Using the standard - normal table, \( P(\bar{X}<67.1)=P(Z < 3.21) \). Looking up \( z = 3.21 \) in the standard - normal table, \( P(Z<3.21)=0.9993 \).

Answer:

The probability of randomly selecting 1 woman with a height less than 67.1 inches is \( 0.8238 \).
The probability of selecting a sample of 12 women with a mean height less than 67.1 inches is \( 0.9993 \).
Since \( 0.9993>0.8238 \), it is more likely to select a sample of 12 women with a mean height less than 67.1 inches. So the answer is not option A. (It seems there is an error in option A as a higher probability means more likely).