Final answer:
To determine if there is sufficient evidence that the valve performs above the specifications, we can conduct a hypothesis test using a one-sample t-test. The null hypothesis is that the mean pressure of the valve is equal to the designed mean pressure of 4.5 lbs/square inch. The alternative hypothesis is that the mean pressure is greater than 4.5 lbs/square inch.
Step-by-step explanation:
To determine if there is sufficient evidence that the valve performs above the specifications, we can conduct a hypothesis test. The null hypothesis (H0) is that the mean pressure of the valve is equal to the designed mean pressure of 4.5 lbs/square inch. The alternative hypothesis (Ha) is that the mean pressure is greater than 4.5 lbs/square inch.
Using a one-sample t-test with a significance level of 0.01, we can calculate the test statistic and compare it to the critical value from the t-distribution table. The test statistic can be calculated using the formula: t = (sample mean - hypothesized mean) / (standard deviation / square root of sample size).
Plugging in the values from the question, we have: t = (4.6 - 4.5) / (0.5 / sqrt(180)). Calculating this expression will give us the test statistic. We can then refer to the t-distribution table to find the critical value at the 0.01 significance level.
If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is sufficient evidence that the valve performs above the specifications. If the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis and do not have sufficient evidence to conclude that the valve performs above the specifications.