Mathematics, 08.10.2021 19:30 khuak877

Under normal operations the process that produces hypodermic needles produces needles whose diameters are normally distributed with a mean of 1.65 mm and a standard deviation of 0.01 mm. 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) One day, when the test is conducted, the average diameter of the needles is 1.648 mm. Notice, the machine is not shut down this day. What is the p-value of the test this day?

b) Another day, when the test is conducted the average diameter of the needles is 1.658 mm. Notice, the machine is shut down this day. What is the p-value of the test this day?

c) What is the significance level of the test?

d) When the machine is operating normally, what is the probability it is shut down unnecessarily based on the results of the test? (In other words, what is the probability of

making a Type I error?)

e) 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?

f) 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?

g) There is a screw that loosens on occasion causing the standard deviation of the diameters to increase to 0.02mm (the mean is still 1.65mm). Find the sample size needed to have a 95% chance of detecting this when it has occurred (assuming the rejection region remains the same whatever sample size you use).

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