Let's address the problem step-by-step.
### Part (a): Hypothesis Testing
1. State the hypotheses:
We are to test if the population mean is less than 12.5 psi. This calls for a one-tailed test. The appropriate hypotheses are:
- Null Hypothesis (H₀): [tex]\(\mu = 12.5\)[/tex]
- Alternative Hypothesis (Hₐ): [tex]\(\mu < 12.5\)[/tex]
So, the correct choice is D: [tex]\( H_0: \mu = 12.5 \)[/tex]
2. Find the test statistic:
The test statistic used in this scenario is the t-statistic since we are dealing with the sample data. The calculated t-statistic is:
[tex]\[
t = -22.17
\][/tex]
3. Determine the p-value:
The p-value associated with this t-statistic is:
[tex]\[
p = 0.0
\][/tex]
4. Interpret the results:
- Compare the p-value with the significance level ([tex]\(\alpha = 0.05\)[/tex]):
[tex]\[
\text{Since } p = 0.0 \text{ is less than } 0.05, \text{ we reject } H_0.
\][/tex]
- There is sufficient evidence to conclude that the population mean is less than 12.5 psi at a significance level of 0.05.
- Therefore, using the given measurements, it appears that the Patriots' balls are underinflated.
### Part (b): Confidence Interval Construction
1. Construct a 95% confidence interval for the mean psi for the Patriots' footballs:
The 95% confidence interval for the mean is calculated as follows:
[tex]\[
\text{Confidence Interval} = (11.13, 11.38)
\][/tex]
2. Support for Conclusion in Part (a):
The confidence interval support's the conclusion in part (a) because the entire interval (11.13, 11.38) lies below 12.5 psi. This further affirms that the mean psi of the footballs is significantly less than 12.5 psi, reinforcing the fact that the Patriots' balls were indeed underinflated based on this sample data.
### Summarized Answers:
1. Test Statistic:
[tex]\[
t = -22.17
\][/tex]
2. p-value:
[tex]\[
p = 0.0
\][/tex]
3. Conclusion:
Since the p-value is less than the significance level, we reject [tex]\(H_0\)[/tex]. There is sufficient evidence to conclude that the population mean is less than 12.5 psi at a significance level of 0.05. Using only the given measurements, it appears that the Patriots' balls are underinflated.
4. 95% Confidence Interval:
[tex]\[
(11.13, 11.38) \text{ psi}
\][/tex]
The confidence interval supports the rejection of the null hypothesis because it does not include the value 12.5 psi, thereby affirming that the footballs were underinflated based on the sample measurements.
