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 6.9 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 16 engines and the mean pressure was 7.1 pounds/square inch with a standard deviation of 1.0. A level of significance of 0.05 will be used. Assume the population distribution is approximately normal. Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places Answer How to Enter) 8 Points Keypad Keyboard Shortcuts Reject H, ift > Prev 2020 Mastering

Answer 1

Answer :

The provided sample mean is x̅ =7.1pounds/square inch and the sample standard deviation is s=1.0, and the sample size is n = 16.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H₀ : μ ≤ 6.9

H₁ : μ > 6.9

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is at 0.05 significance level with (16 - 1)= 15 degrees of freedom is t_c =1.753.

The rejection region for this right-tailed test is R={t:t>1.753}

That is,

Reject H₀ if t > 1.753

(3) Test Statistics

The t-statistic is computed as follows:

t= (x̅ −μ0)/(s/√n)

=(​7.1−6.9​)/(1.0/√16)

=0.2/0.25

=0.8

(4) Decision about the null hypothesis

Since it is observed that t=0.8 ≤ tc​=1.753, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value for t score= 0.8 with (16 - 1) = 15 degrees of freedom for right tailed test is p=0.2181, and since p = 0.2181 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis H₀ is not rejected. Therefore, there is not enough evidence to claim that the value perform the above the specification (that is mean pressure is 6.9 pounds/square inch) its means that the population mean μ is greater than 6.9, at the 0.05 significance level.