Question

Under normal operations the process that produces hypodermic needles produces needles whose diameter has a mean of 1.65mm and a standard deviation of 0.01mm. Every day, before operations begin, the machine is tested to make sure it is operating normally. The test consists of measuring the lengths of 5 needles coming off the machine. If the average diameter of the 5 needles is either greater than 1.655 or less than 1.645 the machine is shut down and inspected.

a. When the machine is operating normally, what is the probability it is shut down unnecessarily based on the results of the test?

b. One day, the machine is out of adjustment and is producing needles whose diameters have a mean of 1.655mm and a standard deviation of 0.01mm. What is the power of the test?

c. Another day, the machine is out of adjustment and is producing needles whose diameters have a mean of 1.65mm but a standard deviation of 0.02mm. What is the power of the test?

Answer 1

We know that sampling distribution of sample, mean follow Normal with mean = (population mean)

and standard error =

then ,

(a) When the machine is operating normally , = 1.65 , =0.01

To find out probability of shutting down unnecessarily when the machine is operating normally

To find the probability of shutting down unnecessarily when = 1.65 , =0.01

This is probability of Type I error

We shut down when < 1.645 or > 1.655

That is to find

P( < 1.645 or > 1.655 I = 1.65 , =0.01)

= P( z < -1.12 )+ P(z>1.12)

= 0.1314+0.1314 (from z table)

=0.2628

Therefore ,probability of shutting down unnecessarily when the machine is operating normally =0.2628

b) Power of the test is the probability of shutting down , when the machine is not operating normally

= P( < 1.645 or >1.655 I =1.655,=0.01 )

= P( z < -2.24 )+ P(z>0)

= 0.0125+0.5    (from z table)

=0.5125

Power of the test =0.5125

c)

Power of the test is the probability of shutting down , when the machine is not operating normally

= P( < 1.645 or >1.655 I =1.65,=0.02 )

= P( z < -0.56 )+ P(z>0.56)

= 0.2877+0.2877    (from z table)

=0.5755

Power of the test =0.5755