A manufacturer has developed a new drive belt for a machine. The
original drive belts are known to have a population mean (average)
operating life of 3500 hours. We obtain the lifetimes a random sample
of 50 of the new drive belts, resulting in a sample average of 3675
hours with a sample standard deviation of 500.
a) Obtain a 98% confidence interval for the population average with
the new drive belt.
b) From (a), can one conclude with 98% confidence that the population
average is above 500, i.e. that the new drive belt lasts longer (on
the average) the the orginal drive belt?
c) What is the maximum confidence at which we can conclude that the
new drive belt lasts longer?
d) Suppose that we wish to operate at a margin of error of 75, still
maintaining 98% confidence. What is the sample size needed to achieve
this?
e) Suppose our desire is to establish that the new drive belt lasts
longer: Run the appropriate hypothesis test using a type I error
probability = .01 Specifically, (i) State the Research hypothesis.
              (ii) Obtain the value of the observed Z statistic.
              (iii) Write down the DECISION RULE for deciding whether
or not one can establish the research hypothesis. Give the rule in
terms of the observed Z statistic, and also in terms of the observed
sample average.
              (iv) State the final decision within the context of the problem.
f) Obtain the P-value for the observed value in (e).
g) Explain what type I error means within the context of the problem.
h) Evaluate the type II error probability at the (research hypothesis)
population average value of 3675.
i) Explain what the type II error means within the context of the problem.
j) Suppose that our desire is to achieve a power of .99 at the
population average value of 3675 (to go along with type I error of
probability of .01). What is the total sample size needed?
k) Explain what 'power' means and why is it that the higher its value the better.
l) Suppose our desire is to establish that the new drive belt lasts
longer. But now we operate with a sample size of 300. Obtain the
appropriate decision rule so that the type I error probability = type
II error probability (evaluated at the population average value of
3600). What is the common value of these two probabilities?