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 6.3 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 130 engines and the mean pressure was 6.5 pounds/square inch. Assume the standard deviation is known to be 0.8. A level of significance of 0.02 will be used. Determine the decision rule. Enter the decision rule.

Answer 1

We conclude that the valve performs above the specifications.

Explanation:

We are given the following in the question:

Population mean, μ = 6.3 pounds per square inch

Sample mean,
`\bar{x}` = 6.5 pounds per square inch

Sample size, n = 130

Alpha, α = 0.02

Population standard deviation, σ = 0.8

First, we design the null and the alternate hypothesis

`H_(0): \mu = 6.3\text{ pounds per square inch}\\H_A: \mu > 6.3\text{ pounds per square inch}`

We use one-tailed z test to perform this hypothesis.

Formula:

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

Putting all the values, we have

`z_(stat) = \displaystyle(6.5 - 6.3)/((0.8)/(√(130)) ) = 2.85`

Now,
`z_(critical) \text{ at 0.02 level of significance } = 2.05`

Decision rule:

If the calculated statistic is greater than the the critical value, we reject the null hypothesis and if the calculated statistic is lower than the the critical value, we accept the null hypothesis

Since,

`z_(stat) > z_(critical)`

We fail to accept the null hypothesis and reject the null hypothesis. We accept the alternate hypothesis.

Thus, we conclude that the valve performs above the specifications.