Question

An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 120 engines and the mean pressure was 5.4 lbs/square inch. Assume the standard deviation is known to be 0.8. If the valve was designed to produce a mean pressure of 5.6 lbs/square inch, is there sufficient evidence at the 0.02 level that the valve does not perform to the specifications? State the null and alternative hypotheses for the above scenario.

Answer 1

Answer:

We conclude that the valve does not perform to the specifications.

Step-by-step explanation:

We are given that the valve was tested on 120 engines and the mean pressure was 5.4 lbs/square inch. Assume the standard deviation is known to be 0.8.

If the valve was designed to produce a mean pressure of 5.6 lbs/square inch.

Let $\mu$ = mean water pressure of the valve.

So, Null Hypothesis, $H_0$ : $\mu$ = 5.6 lbs/square inch     {means that the valve perform to the specifications}

Alternate Hypothesis, $H_A$ : $\mu$ $\neq$ 5.6 lbs/square inch     {means that the valve does not perform to the specifications}

The test statistics that would be used here One-sample z test statistics as we know about the population standard deviation;

                         T.S. =  $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$  ~ N(0,1)

where, $\bar X$ = sample mean pressure = 5.4 lbs/square inch

            $\sigma$ = population standard deviation = 0.8 5.4 lbs/square inch

            n = sample of engines = 120

So, the test statistics  =  $\frac{5.4-5.6}{\frac{0.8}{\sqrt{120} } }$

                                     =  -2.738

The value of z test statistics is -2.738.

Now, at 0.02 significance level the z table gives critical values of -2.3263 and 2.3263 for two-tailed test.

Since our test statistic doesn't lie within the range of critical values of z, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the valve does not perform to the specifications.