Question

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 5.4 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 9 engines and the mean pressure was 5.7 pounds/square inch with a variance of 0.81. A level of significance of 0.025 will be used. Assume the population distribution is approximately normal. Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places.

Answer 1

Answer with explanation:

We assume that the population distribution is approximately normal.

Let $\mu$ be the population mean.

By considering the given information , we have

$H_0:\mu=5.4\\\\H_a:\mu>5.4$, since the alternative hypothesis is right tailed , so the test is right tail test.

Given : Sample size : n=9, which is a small sample (n<30), so we use t-test.

Sample mean : $\overline{x}=5.7$

Variance =$\sigma^2=0.81$

Then, Standard deviation : $\sigma=0.9$

Test statistic :-

$t=\dfrac{\overlien{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

i.e. $t=\dfrac{5.7-5.4}{\dfrac{0.9}{\sqrt{9}}}=1$

Critical value of t=$t_{9-1,0.025}=t_{8,0.025}=2.306$

Since the observed t-value is less than the critical value (1<2.306), so we do not reject the null hypothesis.

Thus , we conclude that we have evidence to do not reject the null hypothesis.