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 5.6 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 160 engines and the mean pressure was 5.7 pounds/square inch. Assume the variance is known to be 0.25. A level of significance of 0.01 will be used. Make a decision to reject or fail to reject the null hypothesis.

Answer 1

The decision rule is

Reject the null hypothesis

Explanation:

From the question we are told that

The population mean is
`\mu = 5.6 \ pounds/inch^2`

The sample size is n = 160

The sample mean is
`\= x = 5.7 \ pounds/ inch^2`

The variance is
`\sigma ^2 = 0.25`

The level of significance is
`\alpha = 0.01`

The null hypothesis is
`H_o : \mu = 5.6`

The alternative hypothesis is
`H_a : \mu > 5.6`

Generally the standard deviation is mathematically represented as

`\sigma = √(\sigma ^2 )`

=>
`\sigma = √(0.25 )`

=>
`\sigma = 0.5`

Generally the test statistics is mathematically represented as

`z = (\= x - \mu )/( (\sigma)/(√(n) ) )`

=>
`z = (5.7 - 5.6 )/( (0.5 )/(√( 160 ) ) )`

=>
`z = 2.53`

From the z table the area under the normal curve to the left corresponding to 2.53 is

`p-value = P(Z > 2.53 ) =0.0057`

From the value obtained we see that the
`p-value < \alpha` hence

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to conclude that the believe that the valve performs above the specifications is true