T-test vs Z-test

Question

It is known that the diameter of a particular type of automobile tyre follows a normal distribution with mean and variance being $600$ millimetres and $3$ millimetres respectively. Ten samples are collected from a large population of manufactured tyres as : It is believed that the manufactured lot has a diameter different from $600$ mm. Which hypotheses test can be used here? What conclusion can be drawn on $H_0$.

Select one:

1. T-test, Failed to reject $H_0$

2. Z-test, Failed to reject $H_0$

3. Z-test, Reject $H_0$

4. F-test, Reject $H_0$

Note: I'm confused between T-test and Z-test. When one should go for T-tset and when for Z test. In this particular senario which one is best and why?

Answer 1

You are given the true mean and variance of the tire diameter. As such, you would use a $Z$-test because the test statistic does not need you to estimate the population variance from the sample.

The $F$-test is not appropriate because that is a test of variance, but the hypothesis concerns the mean of the sample. The $T$-test can be used, in the sense that the testing procedure is statistically valid, but is not the test of first choice here, because it is not as powerful as a $Z$-test when you know the variance of the population.

Answer 2

@Heropup (+1) is correct. Suppose the data were as entered into a recent release of Minitab, as follows:

 595.0   598.5   599.6   603.2   
 601.5   584.6   604.0   597.5   598.1

Then results of the appropriate one-sample z test are as follows:

One-Sample Z: x 

Test of μ = 600 vs ≠ 600
The assumed standard deviation = 3


Variable  N    Mean  StDev  SE Mean       95% CI           Z      P
x         9  598.00   5.78     1.00  (596.04, 599.96)  -2.00  0.046

You reject this two-sided test at the 5% level.

You should find the sample mean $\bar X$ of your data and compute the $Z$ statistic by hand, for an exact answer.