A machine is set to produce disc plates with a mean diameter of 14 mm. A sample of 8 discs gave a mean diameter, x̄ = 14.9 mm, and a standard deviation, s = 1.33 mm. A test was carried out at the 5% level of significance to determine whether the machine is in good working order. Assume that the diameter of the disc follows a normal distribution.
i. State, in symbols, the null and alternate hypotheses for this test.
H0: μ = 14 (null hypothesis)
Ha: μ ≠ 14 (alternate hypothesis)
ii. State, with reasons, whether a t-test or a z-test will be appropriate.
A t-test will be appropriate because the population standard deviation is unknown and the sample size is small (n < 30).
iii. Determine the rejection region(s) of the test.
Since the test is two-tailed and the significance level is 5%, the rejection region(s) will be in both tails of the distribution. The critical t-value for a two-tailed test with 7 degrees of freedom (n-1) and a significance level of 0.05 is approximately ±2.365.
iv. Calculate the value of the test statistic.
The test statistic can be calculated using the formula:
t = (x̄ - μ) / (s / √n)
t = (14.9 - 14) / (1.33 / √8)
t ≈ 2.40
v. State, with reason, a valid conclusion for the test.
Since the calculated test statistic (t ≈ 2.40) falls outside the rejection region (±2.365), we reject the null hypothesis. Therefore, there is sufficient evidence to conclude that the machine is not in good working order.