Question

In order to test whether camshafts are being manufactured to specification a sample of n = 50 camshafts are selected at random. The average value of the sample is calculated to be 4.38 mm and the depths of the camshafts in the sample vary by a standard deviation of s = 0.42 mm.
Test the hypotheses selected previously, by filling in the blanks in the following:

• An estimate of the population mean is .
• The standard error is .
• The distribution is  (examples: normal / t12 / chisquare4 / F5,6).

The test statistic has value TS=  .
Testing at significance level α = 0.01, the rejection region is:
less than  and greater than  (2 dec places).
Since the test statistic  (is in/is not in) the rejection region, there (is evidence/is no evidence) to reject the null hypothesis, H ₀.
There  (is sufficient/is insufficient) evidence to suggest that the average hardness depth, μ, is different to 4.5 mm.

Were any assumptions required in order for this inference to be valid?
a: No - the Central Limit Theorem applies, which states the sampling distribution is normal for any population distribution.
b: Yes - the population distribution must be normally distributed.
Insert your choice (a or b): .

Answer 1

Since the sample size > 30, we can use z- score even though the population standard deviation is not known

An estimate of the population mean is the sample mean = 4.38

Standard error = σ/√n = 0.42/√50 = 0.0594

The complete hypothesis test is given below:

Data:    

n = 50   

μ = 4.5   

σ = 0.42   

x-bar = 4.38   

Hypotheses:   

Ho: μ = 4.5   

Ha: μ ≠ 4.5   

Decision Rule:   

α = 0.01   

Lower Critical z- score = -2.575829304

Upper Critical z- score = 2.575829304

Reject Ho if |z| > 2.58  

Test Statistic:   

SE = s/√n = 0.42/√50 = 0.05939697

z = (x-bar - μ)/SE = (4.38 - 4.5)/0.05939696961967 = -2.020305089

p- value = 0.043351751

Decision (in terms of the hypotheses):

Since 2.020305089 < 2.575829304 we are in the rejection region. So we fail to reject Ho.

Conclusion (in terms of the problem):

There is no sufficient evidence that the mean depth is different from 4.5 mm

The central limit theorem applies since the sample size > 30