For
each problem, state hypotheses and perform the appropriate test. If the level of significance is not given,
just report the p-value and explain what it tells you.

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1.
The
brochure for a Canadian fishing lodge advertises that 75% of its guests catch
fish weighing over 20 pounds. Suppose
that last summer 64 out of a random sample of 83 guests did in fact catch fish
weighing over 20 pounds. Does this
indicate that the population proportion of guests who catch fish weighting over
20 pounds is different from 75% at the 0.05 level of significance?

H_{0}: p = 0.75, H_{A}: p¹ 0.75. Test statistic z = 0.44; the p-value is
2*P(Z>0.44) = 0.657. (the p-value is
doubled because this is a two-sided test.)
Do not reject the null hypothesis at α = 0.05 since the p-value is
so large. At α = 0.05 the evidence
does not indicate that the proportion of guests catching fish weighing over 20
lbs. is different from 75%.

2.
A
random sample of 68 adult coyotes showed the average age to be _{}= 2.05 years. However it is thought that the overall
population mean age of coyotes is μ = 1.75 years. Assume the population standard deviation is
0.82 years, based on past research.
Does the sample data indicate that coyotes in that region tend to live
longer than the population average of 1.75 years? Use α = 0.01.

3.
The
red blood cell count (RBC) for a person is measured in millions of cells per
cubic centimeter of whole blood. This
measure for healthy female adults has a normal distribution with a mean of
about 4.8 million per cc. Suppose a
female patient has taken six blood tests over the past several months and the
results are:

3.5 4.2 4.5 4.6 3.7 3.9

H_{0}: μ = 4.8, H_{A}:
μ < 4.8. Test statistic t = -
4.07; the p-value = P(T<-4.07) = .0048.
We reject the null hypothesis at α = 0.05 since the p-value is less
than 0.05. At this level of
significance, we have evidence that the RBC for this patient is below the
average.

4.
The
Boston Globe reported that the average monthly rent for a two-bedroom apartment
in the greater Boston area was $1650 in summer of 2001. Historically, the standard deviation of the
distribution of these rental amounts has been steady at $75, and the rents have
been approximately normally distributed.
A random sample of 30 rents from last week had a mean of $1607. Is there evidence that the average rent for
a two-bedroom apartment in the Boston area has dropped since summer 2001?

H_{0}: μ = 1650, H_{A}:
μ < 1650. The test statistic is
z = -3.14; p-value = P(Z<-3.14) = 0.00084.
Since the p-value is so small, we conclude our data provides strong
evidence that the average rent has dropped.

5.
The
*Wall Street Journal* reported that 24% of U.S. office workers prefer
“flex time.” A random sample of 66 IBM
employees showed that 23 prefer a flextime schedule. Does this indicate that the proportion of all IBM employees that
prefer flextime is more than 24%?

H_{0}: p=0.24, H_{A}: p
> 0.24. The test statistic is
z=2.06; the p-value = 0.02. This
indicates *possible* evidence that more than 24% prefer flextime; if α=0.10
or 0.05 we would reject H_{0}, but if α=0.01 we would not. Since the p-value is below 0.05 there is
moderate evidence against H_{0 }but the evidence is not extremely
strong since it’s above 0.01.

6.
A
machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7 oz. A random sample of 10 cups from this machine
were obtained and measured with the following results:

6.8 7.2 6.9 7.0 6.9 6.7 7.1 6.9 6.8 7.0

Is there evidence at
α = 0.05 that the machine has slipped out of adjustment and is dispensing
an amount of coffee different from 7.0 oz?

H_{0}: μ = 7, H_{A}: μ¹ 7. We must assume the population is
approximately normal in order to use a t-test.
Test statistic t = -1.48; the p-value = 2*P(T<-1.48) = 2*0.0865 =
0.173. We do not reject H_{0}
since the p-value is greater than α = 0.05. At α = 0.05 there is not any evidence that the average
amount of coffee per cup is different from 7 oz.