Sure, let's go through the solution step-by-step.
1. Given values:
- Sample Size ([tex]\(n\)[/tex]): 21
- Population Mean ([tex]\(\mu\)[/tex]): 65.2 inches
- Sample Mean ([tex]\(\bar{x}\)[/tex]): 57.2 inches (which is 65.2 inches - 8 inches)
- Population Standard Deviation ([tex]\(\sigma\)[/tex]): 13 inches (assumed)
2. Step 1: Calculate the Standard Error (SE):
The standard error of the mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the values:
[tex]\[
SE = \frac{13}{\sqrt{21}} \approx 2.8368
\][/tex]
3. Step 2: Calculate the Z-score:
The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula:
[tex]\[
Z = \frac{\bar{x} - \mu}{SE}
\][/tex]
Substituting the values we have:
[tex]\[
Z = \frac{57.2 - 65.2}{2.8368} \approx -2.8200
\][/tex]
4. Step 3: Determine the Probability:
The probability of a Z-score less than or equal to a given value can be found using the cumulative distribution function (CDF).
Using the Z-score calculated:
[tex]\[
P(Z \leq -2.8200) \approx 0.0024
\][/tex]
So, the probability of selecting a sample of 21 women with a mean height less than 65.2 inches, when the mean is reduced by 8 inches, is approximately 0.0024 (rounded to four decimal places).
