To determine the test statistic for the claim that the population mean IQ of statistic instructors is greater than 110, we can use the t-test formula for the mean. The formula for the t-statistic is as follows:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Where:
- \(\bar{x}\) = Sample mean
- \(\mu_0\) = Population mean under the null hypothesis
- \(s\) = Sample standard deviation
- \(n\) = Sample size
Given:
- Sample mean (\(\bar{x}\)) = 118
- Hypothesized population mean (\(\mu_0\)) = 110
- Sample standard deviation (\(s\)) = 11
- Sample size (\(n\)) = 15
Now we can plug in the values into the formula:
- Calculate \(s / \sqrt{n}\):
\[ s / \sqrt{n} = 11 / \sqrt{15} \]
First, calculate \(\sqrt{15}\):
\[ \sqrt{15} \approx 3.872 \]
Now, calculate \(s / \sqrt{n}\):
\[ s / \sqrt{n} \approx 11 / 3.872 \approx 2.840 \]
- Plug this value into the t-statistic formula:
\[ t = \frac{118 - 110}{2.840} \approx \frac{8}{2.840} \approx 2.816 \]
Thus, the calculated t-statistic is approximately:
\[ t \approx 2.816 \]
Comparing this result with the provided options:
A) t = 9.342
B) t = 2.817
C) t = 3.767
D) t = 10.909
The closest answer, rounding to three decimal places, suggests that the correct answer is:
B) t = 2.817.