Question

steel factory produces iron rods that are supposed to
be 36 inches long. The machine that makes

these rods does not produce each rod exactly 36 inches long. The
lengths of these rods vary slightly.

It is known that when the machine is working properly, the mean
length of the rods is 36 inches.

According to design, the standard deviation of the lengths of all
rods produced on this machine is

always equal to .05 inches. The quality control department at the
factory takes a sample of 40 such

rods each week, calculates the mean length of these rods, and tests
the null hypothesis µ = 36 inches

against the alternative hypothesis µ 6= 36 inches using 1%
significance level. If the null hypothesis

is rejected, the machine is stopped and adjusted. A recent sample
of 40 such rods produced a mean

length of 36.015 inches. Answer the following questions.

(a) Write down the null and alternative hypotheses. Is the
alternative one-sided or two-sided?

(b) State the Type I and Type II errors for this problem and tell
why Type I error is more serious.

(c) State the Z-test statistic for this problem and tell its
distribution.

(d) Is the small, large or both small and large value of the Z-test
statistic that will lead us to reject

the null hypothesis?

(e) State the wanted significance level, determine the critical
value for the rejection region and

write down the decision rule.

(f) Calculate the Z-test statistic from the available data.

(g) Determine whether or not the null hypothesis is rejected at the
significance level wanted and

tell why.

(h) Calculate the P-value and do the hypothesis testing based on
the P-value.

(i) Express the conclusion in the context of the problem, using
common English.

Answer #1

Lazurus Steel Corporation produces iron rods that are supposed
to be 36 inches long. The machine that makes these rods does not
produce each rod exactly 36 inches long. The lengths of the rods
are normally distributed and vary slightly. It is known that when
the machine is working properly, the mean length of the rods is 36
inches. The standard deviation of the lengths of all rods produced
on this machine is always equal to 0.035 inch. The quality...

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