First, let's outline the problem and the hypothesis test setup:
1. Given Data:
- Sample size ([tex]\( n \)[/tex]) = 23
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 226,450 miles
- Population mean (hypothesized) ([tex]\( \mu \)[/tex]) = 220,000 miles
- Sample standard deviation ([tex]\( s \)[/tex]) = 11,500 miles
- Significance level ([tex]\( \alpha \)[/tex]) = 0.01
2. Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\( \mu \leq 220,000 \)[/tex] miles
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): [tex]\( \mu > 220,000 \)[/tex] miles
3. Test Type:
- Because we are testing if the mean lifetime is greater than 220,000 miles, this is a one-tailed test.
4. Test Statistic:
Since the population standard deviation is not known and the sample size is small (n < 30), we use the t-statistic:
[tex]\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\][/tex]
Plugging in the values:
[tex]\[
t = \frac{226450 - 220000}{\frac{11500}{\sqrt{23}}} \approx 2.6898
\][/tex]
5. Degrees of Freedom:
The degrees of freedom [tex]\( (df) \)[/tex] is [tex]\( n - 1 \)[/tex]:
[tex]\[
df = 23 - 1 = 22
\][/tex]
6. Critical Value:
For [tex]\( \alpha = 0.01 \)[/tex] and a one-tailed test with 22 degrees of freedom, the critical value can be found using statistical tables or software. It was found to be:
[tex]\[
t_{critical} = 2.508
\][/tex]
7. P-Value:
The p-value corresponds to the probability of obtaining a test statistic as extreme as the one observed (or more extreme) under the null hypothesis. Here, the p-value is computed as:
[tex]\[
p = 0.0067
\][/tex]
8. Decision Rule:
- If the test statistic is greater than the critical value, then reject the null hypothesis.
- [tex]\( 2.6898 \)[/tex] (test statistic) > [tex]\( 2.508 \)[/tex] (critical value)
9. Conclusion:
Since our test statistic [tex]\( t = 2.6898 \)[/tex] is greater than the critical value [tex]\( t = 2.508 \)[/tex], we reject the null hypothesis.
Therefore, the correct selections are:
- Test statistic: [tex]\( t = 2.6898 \)[/tex]
- Critical value: [tex]\( t = 2.508 \)[/tex]
By rejecting the null hypothesis, we conclude that there is sufficient evidence at the 0.01 significance level to support the claim that the mean lifetime of the car engines is greater than 220,000 miles.
Following the given choices, select:
- Test statistic: [tex]\( t = 2.6898 \)[/tex]
- Critical value: [tex]\( t = 2.508 \)[/tex]
