Sure, let's calculate the test statistic step-by-step for the given hypothesis test.
1. State the given parameters:
- Null hypothesis mean ([tex]\(\mu_0\)[/tex]): [tex]\(52.4\)[/tex]
- Sample mean ([tex]\(\bar{x}\)[/tex]): [tex]\(50.6\)[/tex]
- Population standard deviation ([tex]\(\sigma\)[/tex]): [tex]\(4.38\)[/tex]
- Sample size ([tex]\(n\)[/tex]): [tex]\(40\)[/tex]
2. Calculate the standard error of the mean (SE):
[tex]\[
\text{SE} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\text{SE} = \frac{4.38}{\sqrt{40}}
\][/tex]
Calculate the square root of the sample size first:
[tex]\[
\sqrt{40} \approx 6.3246
\][/tex]
Now, divide the population standard deviation by this value:
[tex]\[
\text{SE} = \frac{4.38}{6.3246} \approx 0.6920
\][/tex]
3. Calculate the test statistic [tex]\(z\)[/tex]:
[tex]\[
z = \frac{\bar{x} - \mu_0}{\text{SE}}
\][/tex]
Substituting the sample mean, null hypothesis mean, and the standard error:
[tex]\[
z = \frac{50.6 - 52.4}{0.6920}
\][/tex]
Calculate the difference in the numerator first:
[tex]\[
50.6 - 52.4 = -1.8
\][/tex]
Now, divide by the standard error:
[tex]\[
z = \frac{-1.8}{0.6920} \approx -2.5991
\][/tex]
Therefore, the test statistic [tex]\(z\)[/tex] is:
[tex]\[
z \approx -2.5991
\][/tex]
So, the test statistic is [tex]\(-2.5991\)[/tex] when rounded to 4 decimal places.
