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: \(\boxed{\text{Fail to Reject Null Hypothesis}}\)