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In the 2015 AFC championship game, there was a charge that the New England Patriots underinflated their footballs for an advantage. The balls should be inflated to between 12.5 and 13.5 pounds per square inch. The accompanying data consists of eleven of the twenty-two measurements on the Patriots' footballs.

[tex]\[
\begin{array}{lllllllllll}
11.45 & 10.95 & 10.90 & 10.90 & 12.30 & 11.80 & 11.95 & 10.50 & 11.85 & 11.50 & 11.55
\end{array}
\][/tex]

a. Test the hypothesis that the population mean is less than 12.5 psi using a significance level of 0.05. State clearly whether the Patriots' footballs are underinflated or not. Assume the conditions for a hypothesis test are satisfied.

Determine the null and alternative hypotheses. Choose the correct answer below.

A. [tex]\(H_0: \mu=12.5\)[/tex]
B. [tex]\(H_0: \mu \neq 12.5\)[/tex]
C. [tex]\(H_0: \mu\ \textgreater \ 12.5\)[/tex] [tex]\(H_a: \mu \neq 12.5\)[/tex] [tex]\(H_a: \mu=12.5\)[/tex] [tex]\(H_a: \mu \leq 12.5\)[/tex]
D. [tex]\(H_0: \mu=12.5\)[/tex]
E. [tex]\(H_0: \mu=12.5\)[/tex]
F. [tex]\(H_0: \mu\ \textless \ 12.5\)[/tex] [tex]\(H_a: \mu\ \textless \ 12.5\)[/tex] [tex]\(H_a: \mu\ \textgreater \ 12.5\)[/tex] [tex]\(H_a: \mu \geq 12.5\)[/tex]

Find the test statistic.

[tex]\[
t = \square
\][/tex]

(Round to two decimal places as needed.)

The p-value is [tex]\(\square\)[/tex].

(Round to three decimal places as needed.)

Interpret the results of the test.



Answer :

GinnyAnswer
Let's go through the complete solution step-by-step for the given problem.

### Step 1: State the Hypotheses
We want to test if the mean pressure of the footballs is less than 12.5 psi.

- Null hypothesis ([tex]\( H_0 \)[/tex]): [tex]\(\mu = 12.5 \)[/tex] psi
- Alternative hypothesis ([tex]\( H_a \)[/tex]): [tex]\(\mu < 12.5 \)[/tex] psi

The correct answer regarding the hypotheses is:
A. [tex]\( H_0: \mu=12.5 \)[/tex]
[tex]\( H_a: \mu<12.5 \)[/tex]

### Step 2: Calculate the Sample Mean and Sample Standard Deviation

Given the measurements of the football pressures:
[tex]\[ 11.45, 10.95, 10.90, 10.90, 12.30, 11.80, 11.95, 10.50, 11.85, 11.50, 11.55 \][/tex]

The sample mean ([tex]\( \bar{x} \)[/tex]) and the sample standard deviation ([tex]\( s \)[/tex]) are calculated as follows:
[tex]\[ \bar{x} = 11.42 \][/tex]
[tex]\[ s = 0.55 \][/tex]

### Step 3: Calculate the Test Statistic

The test statistic for a t-test is given by:
[tex]\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \][/tex]
Where:
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu \)[/tex] is the population mean under the null hypothesis.
- [tex]\( s \)[/tex] is the sample standard deviation.
- [tex]\( n \)[/tex] is the sample size.

Substituting the known values:
[tex]\[ \bar{x} = 11.42 \][/tex]
[tex]\[ \mu = 12.5 \][/tex]
[tex]\[ s = 0.55 \][/tex]
[tex]\[ n = 11 \][/tex]

The test statistic can be calculated as:
[tex]\[ t = \frac{11.42 - 12.5}{0.55 / \sqrt{11}} = -6.51 \][/tex]

### Step 4: Calculate the p-value

Using the t-distribution with [tex]\( n-1 = 10 \)[/tex] degrees of freedom, we find the p-value associated with the test statistic [tex]\( t \)[/tex].

[tex]\[ t = -6.51 \][/tex]

The p-value for [tex]\( t = -6.51 \)[/tex] with 10 degrees of freedom is approximately [tex]\( 0.000 \)[/tex] (rounded to three decimal places).

### Step 5: Interpret the Results

Since the p-value [tex]\( 0.000 \)[/tex] is less than the significance level of [tex]\( 0.05 \)[/tex], we reject the null hypothesis.

### Conclusion

We reject the null hypothesis. Therefore, there is sufficient evidence to conclude that the Patriots' footballs are underinflated.

#### Summary

- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\(\mu = 12.5 \)[/tex]
- Alternative Hypothesis ([tex]\( H_a \)[/tex]): [tex]\(\mu < 12.5 \)[/tex]
- Test Statistic ([tex]\( t \)[/tex]): -6.51
- p-value: 0.000

Interpretation: Reject the null hypothesis. The Patriots' footballs are underinflated.

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