The correct answer is 'Do not reject the null hypothesis as there is not sufficient evidence that the true diameter differs from 0.5 in.'
When the sample size is large and the standard deviation is known, the z-test is typically used.
The T-test is more flexible than the Z-test since the Z-test frequently needs particular circumstances in order to be reliable.
The t test is applied if the population standard deviation is unknown.
Test statistic for one sample t-test is, t = (X‾ - μ0) / (s / √n)
The formula of degrees of freedom for test is, df = n-1 here, n is sample size.
Decision rule:
1. If the P-value is less than the level of significance, the null hypothesis should be rejected; otherwise, it should not be.
2.The null hypothesis is rejected if the t-test statistic value is greater than the t tabulated value; otherwise, the null hypothesis is not rejected.
(a) From the given information, n=15, t=1.66, α =0.05 we can derive
degrees of freedom = df= n-1 which is 15-1 = 14
Thus, the critical value is, 2.145.
Now, since the test statistic value is less than the critical value, do not reject the null hypothesis.
Here, the test statistic value do not lie in the critical region. The null hypothesis cannot be rejected. There is insufficient evidence that the true diameter differs from 0.5 in.
(b) From the given information, n=15, t = -1.66, α =0.05 we can derive
degrees of freedom = df= n-1 which is 15-1 = 14
Thus, the critical value is, -2.145.
Now, since the test statistic value does not lie in the rejection region, do not reject the null hypothesis.
Here, the test statistic value does not lie in the critical region. The null hypothesis cannot be rejected. There is insufficient evidence that the true diameter differs from 0.5 in.
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