An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such that it would produce a mean pressure of 49 pounds/square inch. it is believed that the valve performs above the specifications. The valve was tested on 22 engines and the mean pressure was 5.2 pounds/square inch with a standard deviation of 0.7 . A level of significance of 0.025 will be used. Assume the population distribution is approximately normal, Make the decision to reject or fail to reject the null hypothesis.

Answer
Tables
Keypad
Keyboard Shortcuts
Reject Null Hypothesis
Fail to Reject Null Hypothesis

Step 1 ：Define the null hypothesis as the mean pressure of the valve being 49 pounds/square inch, and the alternative hypothesis as the mean pressure being greater than 49 pounds/square inch.

Step 2 ：Given a sample mean of 5.2 pounds/square inch, a sample standard deviation of 0.7, and a sample size of 22.

Step 3 ：Also given a significance level of 0.025.

Step 4 ：Perform a one-sample t-test to test the hypothesis.

Step 5 ：Calculate the test statistic using the formula \(t_{stat} = \frac{x_{bar} - \mu}{s / \sqrt{n}}\), where \(\mu = 49\), \(x_{bar} = 5.2\), \(s = 0.7\), and \(n = 22\). The calculated \(t_{stat}\) is -293.4860146860946.

Step 6 ：Find the critical value \(t_{crit}\) from the t-distribution table with \(n-1\) degrees of freedom and a significance level of 0.025. The \(t_{crit}\) is 2.079613844727662.

Step 7 ：Compare the test statistic with the critical value. Since the test statistic is much less than the critical value, we fail to reject the null hypothesis.

Step 8 ：Conclude that there is not enough evidence to suggest that the valve performs above the specifications.

Step 9 ：Final Answer: \(\boxed{\text{Fail to Reject Null Hypothesis}}\)