Question

Consider a manufacturing process that is producing hypodermic needles that will be used for blood donations. 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, rendering 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. Suppose the most recent random sample of 35 needles have an average diameter of 1.64 mm and a standard deviation of 0.07 mm. Also, suppose the diameters of needles produced by this manufacturing process have a bell shaped distribution.​

Describe what a Type I error would be in this study.​

Answer 1

Type I error will be Rejecting the null hypothesis that the average diameter of needles is 1.65 mm and assume that the average diameter of needles is different from 1.65 mm but the fact is that the null hypothesis was true that the average diameter of needles is 1.65 mm.

Explanation:

We are given that a manufacturing process is producing hypodermic needles that will be used for blood donations. 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, rendering the donated blood useless.

So, Null Hypothesis,
`H_0` :
`\mu` = 1.65 mm

Alternate Hypothesis,
`H_1` :
`\mu\\eq` 1.65 mm

Also, the most recent random sample of 35 needles have an average diameter of 1.64 mm and a standard deviation of 0.07 mm.

Now, Type I error Type I error states that : Probability of rejecting null hypothesis given the fact that null hypothesis was true. It is the the probability of rejecting a true hypothesis.

So, in our question Type I error will be Rejecting the null hypothesis that the average diameter of needles is 1.65 mm and assume that the average diameter of needles is different from 1.65 mm but the fact is that the null hypothesis was true that the average diameter of needles is 1.65 mm.

Answer 2

`\text{Average diameter is 1.65 mm and we decide that it is not 1.65 mm.}`

Explanation:

We are given the following in the question:

The needle size should not be too big and too small.

The diameter of the needle should be 1.65 mm.

We design the null and the alternate hypothesis

`H_(0): \mu = 1.65\text{ mm}\\H_A: \mu \\eq 1.65\text{ mm}`

Sample size, n = 35

Sample mean,
`\bar{x}` = 1.64 mm

Sample standard deviation, s = 0.07 mm

Type I error:

• It is the error of rejecting the null hypothesis when it is true.
• It is also known as false positive error.
• It is the rejecting of a true null hypothesis.

Thus, type I error in this study would mean we reject the null hypothesis that the average diameter is 1.65 mm but actually the average diameters of the needle is 1.65 mm.

Thus, average diameter is 1.65 mm and we decide that it is not 1.65 mm.