Question

2.2.19 Needles! Consider a manufacturing process that is producing hypodermic needles that will be used for blood donantions. These needles need to have a diameter of 1.65 mm - too big and they would hurt the donor (even more than usual), too small and they would rupture the red blood cells, rednering the donated blood useless. Thus, the manufacturing process would have to be closely monitored to detect any significant departures from the desired diameter. During every shift, quality control personnel take a random sample of several needles and measure their diameters. If they discover a problem, they will stop the manufacturing process until it is corrected. For now, suppose that a "problem" is when the sample average diameter turns out to be statistically significantly different from the target of 1.65 mm.

Answer 1

1)

As we are interested in the average diameter of needles produced, so the parameter of interest here is -

The true, long-run average needle diameter ().

----------------------------------------

2)

Now, we are just interested to check if the true mean is significantly different from 1.65 mm. This is because both smaller and larger diameter are harmful.

Thus the hypothesis to be tested must be two tailed given as -

---------------------------------------------

3)

Type 1 error occurs when we reject a true null hypothesis. So, in this case it would occur if -

Average diameter is 1.65 mm, and we don't find strong evidence that it is different than 1.65. mm.

---------------------------------

4)

Type 2 error occurs when you fail to reject a true null hypothesis. So, in this case it occurs if -

Average diameter is not 1.65 mm, but we don't find strong evidence that it is different than 1.65 mm.