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# Statistics

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Exercise 1

One step in the manufacture of a certain metal clamp involves the drilling of four holes. In a sample of 150 clamps, the average time needed to complete this step was 72 seconds and the standard deviation was 10 seconds.

(a) Find the probability that the average time to complete the step is between 65 and 74 seconds (inclusive).
(b) Find a 93% confidence interval for the mean time needed to complete the step.
(c) What is the confidence level of the interval (71, 73)?
(d) How many clamps must be sampled so that a 88% confidence interval specifies the mean to within +-1.5 seconds?
(e) Find a 97% lower confidence bound for the mean time to complete the step.

Exercise 2

A new concrete mix is being designed to provide adequate compressive strength for concrete blocks. The specification for a particular application calls for the blocks to have a mean compressive strength greater than 1350 kPa. A sample of 100 blocks is produced and tested. Their mean compressive strength is 1356 kPa and their standard deviation is 70 kPa.

a) Do you believe it is plausible that the blocks do not meet the specification, or are you convinced that they do? Explain your
reasoning by carrying out the appropriate hypothesis test.
b) If a sample of 10 blocks is produced and tested with their mean compressive strength at 1372 kPa and their standard deviation at 68 kPa. Find the P-value based on the same hypothesis.

Exercise 3

A particular type of gasoline is supposed to have a mean octane rating of 90%. Five measurements are taken of the octane rating and the results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3.

a) Test the requirements of the octane rating using the appropriate hypothesis test (assume alpha = 0.01).
b) Verify your result in part (a) using the appropriate confidence interval.