An auto maker is interested in information about how long transmissions last. A sample of transmissions is run constantly, and the number of miles before the transmission fails is recorded. The auto maker claims that the transmissions can run constantly for over 150,000 miles before failure. The results of the sample are given below.

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|l|}{Miles (1000s of miles)} \\
\hline Mean & [tex]$\overline{150.7}$[/tex] \\
\hline Variance & 4.551 \\
\hline Observations & 42 \\
\hline Hypothesized Mean & 150 \\
\hline df & 41 \\
\hline t Stat & 2.17 \\
\hline [tex]$P(T \le t)$[/tex] one-tail & 0.018 \\
\hline [tex]$t$[/tex] Critical one-tail & 1.683 \\
\hline [tex]$P(T \le t)$[/tex] two-tail & 0.036 \\
\hline [tex]$t$[/tex] Critical two-tail & 2.02 \\
\hline Confidence Level (95\%) & 0.665 \\
\hline
\end{tabular}

[tex]$n = 42$[/tex]

[tex]$\bar{x} = 150.7$[/tex]

Degrees of freedom [tex]$= 41$[/tex]

[tex]$s = 1.234$[/tex]



Answer :

Let's analyze the given problem step by step to determine whether we should accept or reject the null hypothesis using the provided sample data.

### 1. Setting up the Hypotheses

The auto maker claims that the transmissions can last over 150,000 miles before failing.

- Null Hypothesis (\( H_0 \)): The mean number of miles before failure is 150,000 miles or less.
[tex]\[ H_0: \mu \leq 150,000 \][/tex]

- Alternative Hypothesis (\( H_1 \)): The mean number of miles before failure is greater than 150,000 miles.
[tex]\[ H_1: \mu > 150,000 \][/tex]

### 2. Test Statistics and Critical Values

From the given data:
- Sample Mean (\( \bar{x} \)) = 150.7
- Hypothesized Mean (\( \mu_0 \)) = 150
- Sample Size (n) = 42
- Sample Variance = 4.551
- Degrees of Freedom (df) = 41
- t Stat = 2.17
- \( P(T \leq t) \) one-tail = 0.018
- t Critical one-tail = 1.683
- \( P(T \leq t) \) two-tail = 0.036
- t Critical two-tail = 2.02

### 3. One-Tailed Testing Decision

For a one-tailed test, we're interested in determining if the mean number of miles is greater than 150,000. Here are the rules for our decision-making process:

- If \( t \) Stat > t Critical one-tail
- If \( P \) value one-tail < 0.05

Given our data:

- t Stat = 2.17
- t Critical one-tail = 1.683
- \( P(T \leq t) \) one-tail = 0.018

#### Comparison:

- Since 2.17 > 1.683, we can reject the null hypothesis.
- Since 0.018 < 0.05, we can also reject the null hypothesis.

### 4. Two-Tailed Testing Decision

For completeness, let's also consider the two-tailed test, even though the claim was specifically for over 150,000 miles.

For a two-tailed test:

- If \( t \) Stat > t Critical two-tail
- If \( P \) value two-tail < 0.05

Given our data:

- t Stat = 2.17
- t Critical two-tail = 2.02
- \( P(T \leq t) \) two-tail = 0.036

#### Comparison:

- Since 2.17 > 2.02, we can reject the null hypothesis.
- Since 0.036 < 0.05, we can also reject the null hypothesis.

### 5. Conclusion

Based on the above analysis for both one-tailed and two-tailed testing, we have:

#### One-Tailed:
- Decision: Reject the null hypothesis.

#### Two-Tailed:
- Decision: Reject the null hypothesis.

### Final Decision:
The evidence from the sample data supports the claim that the transmissions can indeed last over 150,000 miles before failing. Therefore, based on both one-tailed and two-tailed tests, we reject the null hypothesis.