Question

An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 110 engines and the mean pressure was 4.6 lbs/square inch. Assume the standard deviation is known to be 0.8. If the valve was designed to produce a mean pressure of 4.5 lbs/square inch, is there sufficient evidence at the 0.02 level that the valve does not perform to the specifications?

Answer 1

Answer: NO.

Explanation:

As per given , we have to test the hypothesis.

`H_0:\mu=4.5\\\\ H_a:\mu\\eq4.5`

∵
`H_a` is two-tailed , so our test is a two-tailed test.

Also, the standard deviation is known to be 0.8 , so we use z-test.

Test statistic:
`z=\frac{\overline{x}-\mu}{(\sigma)/(√(n))}`

, where
`\overline{x}` = Sample mean

`\mu` = population mean

`\sigma` = Population standard deviation

n= Sample size

Put
`\overline{x}=4.6`

`\mu=4.5`

`\sigma=0.8`

n= 110 , we get

`z=(4.6-4.5)/((0.8)/(√(110)))\approx1.31`

P-value for two tailed test = 2P(Z>|z|)

= 2P(Z>|1.31|) = 2(1-P(Z<1.31)) [∵ P(Z>z)=1-P(Z<z)]

=2(1- 0.9049) [By z-table]

=0.1902

Decision : ∵ P-value (0.1902) > Significance level (0.02).

It means we do not reject the null hypothesis.

[When P-values < Significance level then we reject the null hypothesis.]

Conclusion : We do not have sufficient evidence at the 0.02 level that the valve does not perform to the specifications.