The diameter of 3.5 centimetre discs is normally distributed.Periodically, quality control inspectors at Winnipeg Discs randomly select a sample of 16 discs.If the mean diameter of the discs is too large or too small the disc punch is shut down for adjustment; otherwise, the punching process continues.The last sample showed a mean and standard deviation of 3.49 and 0.08 centimetres, respectively.Using $\alpha$ = 0.05, the appropriate decision is ___.
A)reject the null hypothesis and shut down the punch
B)reject the null hypothesis and do not shut down the punch
C)do not reject the null hypothesis and shut down the punch
D)do not reject the null hypothesis and do not shut down the punch
E)do nothing

Question

Quizplus · Accepted Answer

The null hypothesis is that the mean diameter of the discs is equal to 3.5 cm (i.e. μ = 3.5).

We can use a t-test for the sample of 16 discs, since the population standard deviation is not known.

The test statistic is calculated as:

t = (sample mean - hypothesized mean) / (sample standard deviation / √n)

t = (3.49 - 3.5) / (0.08 / √16)

t = -1

Using a t-distribution table with 15 degrees of freedom (since n-1 = 16-1 = 15), the critical value for a two-tailed test with α = 0.05 is 2.131.

Since the calculated t-value (-1) does not exceed the critical value (-2.131 and 2.131), we fail to reject the null hypothesis.

Therefore, we do not shut down the punch and continue with the punching process.

The correct choice is D, do not reject the null hypothesis and do not shut down the punch.

The Diameter of 3 α\alphaα = 005, the Appropriate Decision Is ___

The Diameter of 3 $\alpha$ = 005, the Appropriate Decision Is ___