Respuesta :
Answer:
The p-value of the test is 0.0416 < 0.05, which means that the null hypothesis is rejected.
Step-by-step explanation:
The engineer designed the valve such that it would produce a mean pressure of 5.7 pounds/square inch. It is believed that the valve performs above the specifications.
At the null hypothesis, we test that the mean is the specification value of 5.7, that is:
[tex]H_0: \mu = 5.7[/tex]
At the alternate hypothesis, we test that the mean is above the specifications, that is, above 5.7. So:
[tex]H_a: \mu > 5.7[/tex]
The test statistic is:
[tex]t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, s is the standard deviation of the sample and n is the size of the sample.
5.7 is tested at the null hypothesis:
This means that [tex]\mu = 5.7[/tex]
The valve was tested on 8 engines and the mean pressure was 6.2 pounds/square inch with a variance of 0.49.
This means that [tex]n = 8, X = 6.2, s = \sqrt{0.49} = 0.7[/tex]
Test statistic:
[tex]t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{6.2 - 5.7}{\frac{0.7}{\sqrt{8}}}[/tex]
[tex]t = 2.02[/tex]
P-value of the test and decision:
The p-value of the test is the probability of finding a sample mean above 6.2, which is the p-value of t = 2.02, using a right-tailed test with 8 - 1 = 7 degrees of freedom.
With the help of a calculator, this p-value is 0.0416.
The p-value of the test is 0.0416 < 0.05, which means that the null hypothesis is rejected.
