Questions: An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 180 engines and the mean pressure was 4.6 lbs / square inch. Assume the standard deviation is known to be 0.5. If the valve was designed to produce a mean pressure of 4.5 lbs / square inch, is there sufficient evidence at the 0.02 level that the valve performs above the specifications? State the null and alternative hypotheses for the above scenario.

Solution Steps

We define the null and alternative hypotheses as follows:

• Null Hypothesis (\(H_0\)): The mean pressure is equal to the specifications, \( \mu = 4.5 \, \text{lbs/square inch} \).
• Alternative Hypothesis (\(H_a\)): The mean pressure is greater than the specifications, \( \mu > 4.5 \, \text{lbs/square inch} \).

The standard error (\(SE\)) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.5}{\sqrt{180}} \approx 0.0373 \]

The Z-test statistic is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{4.6 - 4.5}{0.0373} \approx 2.6833 \]

For a right-tailed test, the P-value is calculated as: \[ P = 1 - T(Z) \approx 0.0036 \]

We compare the P-value to the significance level (\(\alpha = 0.02\)):

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

Since \(0.0036 < 0.02\), we reject the null hypothesis.

Final Answer