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:
- First, calculate \(\frac{\sigma}{\sqrt{n}}\):
\[ \frac{\sigma}{\sqrt{n}} = \frac{3.44}{\sqrt{219}} \approx \frac{3.44}{14.8} \approx 0.2324 \]
- 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: \(-0.86\)