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 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.

Answer 1

The valves perform above expectations.

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,
`\bar{x}` = 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:

`H_0` : μ = 7.6 pounds per square inch

`H_A`: μ > 7.6 pounds per square inch

Formula:

`z_(stat) = \displaystyle\frac{\bar{x}-\mu}{(\sigma)/(√(n))}`

`z_(stat) = \displaystyle(7.8-7.6)/((1)/(√(140)))`

`z_(stat) = 2.366`

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,
`z_(critical)` = 1.645

Result:

Since,
`z_(stat) > z_(critical)`

`H_(0)` is rejected.

Thus, we accept the alternate hypothesis.

Hence, the valve perform above expectations.