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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 7.6 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 140 engines and the mean pressure was 7.8 pounds/square inch. Assume the standard deviation is known to be 1.0. A level of significance of 0.05 will be used. Make a decision to reject or fail to reject the null hypothesis.

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Answer:

The valves perform above expectations.

Step-by-step explanation:

We are given the following information in the question:

Population mean, μ = 7.6 pounds per square inch

Sample size, n = 140

Sample mean, [tex]\bar{x}[/tex] = 7.8 pounds per square inch

Population standard deviation = 1.0 pounds per square inch

Level of significance = 0.05

We design the null and alternate hypothesis:

[tex]H_0[/tex] :  μ = 7.6 pounds per square inch

[tex]H_A[/tex]:  μ > 7.6 pounds per square inch

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]z_{stat} = \displaystyle\frac{7.8-7.6}{\frac{1}{\sqrt{140}}}[/tex]

[tex]z_{stat} = 2.366[/tex]

Now, we are performing a one tail test with level of significance of 0.05, we calculate the critical value of z with the help of standard normal distribution table.

Thus, [tex]z_{critical}[/tex] = 1.645

Result:

Since, [tex]z_{stat} > z_{critical}[/tex]

[tex]H_{0}[/tex] is rejected.

Thus, we accept the alternate hypothesis.

Hence, the valve perform above expectations.

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