Question

A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the pin head. Measurements on the Rockwell hardness are made on each of the 12 pins, yielding an average value of 48.50 with a sample standard deviation of 1.5. Let µ be the true mean Rockwell hardness on the pin head. (a) Assuming the measurements to be normally distributed, construct a 90 percent confidence interval for µ.

b) Mark as True or False.

(i) Given that you obtained the correct interval in part (a), the probability that µ lies in the interval is 90 percent.

(ii) Given the same data set, a 95 percent confidence interval for µ will be longer.

(iii) There is an independence assumption built into the definition of a random sample.

(iv) If we knew that 1.5 was the true standard deviation, then we would construct a shorter 90 percent confidence interval than the one constructed in part (a)

Answer 1

Let X be the Measurements on the Rockwell hardness.

Data:

n=12

The 90% confidence interval for mean rockwell hardness is given by:

The 90% confidence interval for mean rockwell hardness is

b)

i) False. The 90% confidence interval indicates that we are 90% confident that the true population mean lies in this interval.

ii) True: since the as the confidence level increases the margin of error increases therefore for the same data a 95 percent confidence interval for µ will be longer.

iii) True. The independence of sample is assumed in the definition of a random sample.

iv) True. Since we knew the true standard deviation therefore we used the z critical value =1.64 to construct a 90 percent confidence interval for mean which results in reducing margin of error.