Questions: An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 140 engines and the mean pressure was 6.8 lbs / square inch. Assume the standard deviation is known to be 0.8. If the valve was designed to produce a mean pressure of 6.9 lbs / square inch, is there sufficient evidence at the 0.1 level that the valve does not perform to the specifications? State the null and alternative hypotheses for the above scenario. H0: Ha:

Solution Steps

We define the null and alternative hypotheses as follows:

• Null Hypothesis \( H_0 \): The valve produces a mean pressure of \( 6.9 \, \text{lbs/square inch} \).
• Alternative Hypothesis \( H_a \): The valve does not produce a mean pressure of \( 6.9 \, \text{lbs/square inch} \).

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.8}{\sqrt{140}} \approx 0.0676 \]

The Z-test statistic is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{6.8 - 6.9}{0.0676} \approx -1.479 \]

For a two-tailed test, the P-value is calculated as: \[ P = 2 \times (1 - T(|z|)) \approx 0.1391 \]

We compare the P-value to the significance level \( \alpha = 0.1 \):

• If \( P < \alpha \), we reject the null hypothesis.
• If \( P \geq \alpha \), we fail to reject the null hypothesis.

Since \( P \approx 0.1391 \) is greater than \( \alpha = 0.1 \), we fail to reject the null hypothesis. Therefore, there is not sufficient evidence that the valve does not perform to the specifications.

Final Answer