Math 247: SOLUTIONS

Practice problems for hypothesis tests: Chapters 20, 21, 23

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.

 

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?

 

H0: p = 0.75, HA: 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.

 

H0: μ = 1.75, HA: μ > 1.75.  Test statistic z = 3.33; the p-value = P(Z>3.33) = 0.0004.  Since the p-value is so small, it tells us our evidence against the null hypothesis is very strong.  Reject the null hypothesis at α = 0.01.  At α = 0.01 the data do indicate the average age of coyotes is higher than 1.75 years.

 

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

 

            Do the data indicate that the RBC for this patient is lower than 4.8?  Use α = 0.05

 

H0: μ = 4.8, HA: 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?

 

H0: μ = 1650, HA: 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%?

H0: p=0.24, HA: 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 H0, but if α=0.01 we would not.  Since the p-value is below 0.05 there is moderate evidence against H0 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?

           

      H0: μ = 7, HA: 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 H0 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.