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Sagot :
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.
### 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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