Answer :
To calculate the test statistic for a sample, you'll typically use the formula for the t-statistic, since we are dealing with sample data and not population data.
The formula for the t-statistic is given by:
[tex]t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
Where:
- [tex]\bar{x}[/tex] is the sample mean, which is 39.1.
- [tex]\mu[/tex] is the population mean. If not provided and if you're conducting a hypothesis test, you would use the null hypothesis mean; in practice problems, sometimes a assumed mean might be given.
- [tex]s[/tex] is the sample standard deviation, which is 1.03.
- [tex]n[/tex] is the number of subjects in the sample, which is 21.
If we assume that the population mean [tex]\mu[/tex] we are testing against is, for instance, 40 (since it's not provided), we can substitute these values into the formula:
[tex]t = \frac{39.1 - 40}{\frac{1.03}{\sqrt{21}}}[/tex]
This simplifies to:
[tex]t = \frac{-0.9}{\frac{1.03}{4.5826}}[/tex]
First calculate [tex]\frac{1.03}{4.5826}[/tex]:
[tex]\approx \frac{1.03}{4.5826} \approx 0.224835[/tex]
Now substitute back to find the t-statistic:
[tex]t = \frac{-0.9}{0.224835} \approx -4.00[/tex]
(rounded to two decimal places)
Therefore, the test statistic is approximately [tex]-4.00[/tex]. This represents how many standard deviations away the sample mean is from the hypothesized population mean. If conducting a hypothesis test, you would then compare this test statistic to a critical value from the t-distribution to determine statistical significance.
