Question

National report mean height is 69.4 inches
In a sample of 219 men between 60 and 69, th mean height was 69.2.
Public officials want to determine if the mean height for older men is less than the mean height of all men.
Population standard deviation is 3.44
A=0.01

COMPUTE THE VALUE OF THE TEST STATISTIC

Answer 1

To compute the value of the test statistic, we can use a Z-test for the sample mean since the population standard deviation is known. The formula for the Z-test statistic is:

\[
Z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{X}\) = sample mean
- \(\mu_0\) = population mean (from the national report)
- \(\sigma\) = population standard deviation
- \(n\) = sample size

Given:
- \(\bar{X} = 69.2\) inches (sample mean)
- \(\mu_0 = 69.4\) inches (population mean)
- \(\sigma = 3.44\) inches (population standard deviation)
- \(n = 219\) (sample size)

Now, we can plug in the values:

1. First, calculate \(\frac{\sigma}{\sqrt{n}}\):

\[
\frac{\sigma}{\sqrt{n}} = \frac{3.44}{\sqrt{219}} \approx \frac{3.44}{14.8} \approx 0.2324
\]

2. Then, compute the value of the test statistic \(Z\):

\[
Z = \frac{69.2 - 69.4}{0.2324} = \frac{-0.2}{0.2324} \approx -0.8596
\]

Thus, the value of the test statistic \(Z\) is approximately \(-0.86\).

نتيجة - Z: \(-0.86\)