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.
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.
b. 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 higher probability.
c. it is more likely to select 1 woman with a height less than 67.1 inches because the probability is lower.

Explanation:

Step1: Calculate z - score for a single woman

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
Given \(\mu = 64.6\), \(\sigma=2.7\), and \(x = 67.1\).
\(z=\frac{67.1 - 64.6}{2.7}=\frac{2.5}{2.7}\approx0.93\)
Using the standard normal table, \(P(X\lt67.1)=P(Z\lt0.93)\).
From the standard normal table, \(P(Z\lt0.93)=0.8238\)

Step2: Calculate z - score for the sample mean

The formula for the z - score of the sample mean is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)
Given \(n = 12\), \(\mu = 64.6\), \(\sigma = 2.7\), and \(\bar{x}=67.1\)
\(z=\frac{67.1 - 64.6}{\frac{2.7}{\sqrt{12}}}=\frac{2.5}{\frac{2.7}{3.464}}\)
\(z=\frac{2.5\times3.464}{2.7}=\frac{8.66}{2.7}\approx3.21\)
Using the standard normal table, \(P(\bar{X}\lt67.1)=P(Z\lt3.21)\)
From the standard normal table, \(P(Z\lt3.21)=0.9993\)

Since \(0.9993>0.8238\)

Answer:

B. 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 higher probability.