Test 10B
AP Statistics
Name:
Directions: Work on these sheets. Answer completely, but be concise. A normal probability table is
attached.
1. In formulating hypotheses for a statistical test of significance, the null hypothesis is
often
A) a statement that there is “no effect” or “no difference.”
B) the probability of observing the data you actually obtained.
C) a statement that the data are all 0.
D) 0.05.
2. An agricultural researcher plants 25 plots with a new variety of corn. A 90% confidence
interval for the average yield for these plots is found to be 162.72 ± 4.47 bushels per
acre. Which of the following would produce a confidence interval with a smaller margin
of error than this 90% confidence interval?
A) planting only five plots rather than 25, since five are easier to manage and control.
B) planting 100 plots rather than 25.
C) computing a 99% confidence interval rather than a 90% confidence interval; the increase in
confidence indicates that we have a better interval.
D) none of the above.
3. A small company consists of 25 employees. As a service to the employees, the company arranges
for each to have a complete physical for free. Among other things, the weight of each employee is
measured; the mean weight is found to be 165 pounds. The population standard deviation is 20 pounds.
It is believed that a mean weight of 160
pounds would be normal for this group. To see if there is evidence that the mean weight
of the population of all employees of the company differs significantly from 160, the
hypotheses H0: μ = 160, Ha: μ > 160 are tested. You obtain a P-value of less than 0.1056. Which of
the following is true?
A) At the 5% significance level, you have proved that H0 is true.
B) You have failed to obtain any evidence for Ha.
C) At the 5% significance level, you have failed to prove that H0 is true, and a larger sample
size is needed to do so.
D) None of the above.
4. The mean area μ of the several thousand apartments in a new development is advertised to be 1250
square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an
engineer to measure a sample of apartments to test their suspicion. The appropriate null and alternative
hypotheses, H0 and Ha, for μ
A) are H0: μ = 1250, Ha: μ ¹ 1250.
B) are H0: μ = 1250, Ha: μ < 1250.
C) are H0: μ = 1250, Ha: μ > 1250.
D) cannot be specified without knowing the size of the sample used by the engineer.
5. An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these
plots is J = 150 bushels per acre. Assume that the yield per acre for the new variety of corn follows a
normal distribution with unknown mean μ and that a 95%
confidence interval for μ is found to be 150 ± 3.29. Which of the following is true?
A) A test of the hypotheses H0: μ = 150, Ha: μ ¹ 150 would be rejected at the 0.05 level.
B) A test of the hypotheses H0: μ = 150, Ha: μ > 150 would be rejected at the 0.05 level.
C) A test of the hypotheses H0: μ = 160, Ha: μ ¹ 160 would be rejected at the 0.05 level.
D) No hypothesis test can be conducted because we do not know s.
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Test 10B
6. In tests of significance about an unknown parameter of some population, which of the
following is considered strong evidence against the null hypothesis?
A) The value of an estimate of the unknown parameter based on a simple random
sample from the population is not equal to zero.
B) The value of an estimate of the unknown parameter based on a simple random
sample from the population is equal to zero.
C) We observe a value of an estimate of the unknown parameter based on a simple
random sample from the population that is very consistent with the null hypothesis.
D) We observe a value of an estimate of the unknown parameter based on a simple
random sample from the population that is very unlikely if the null hypothesis is
true.
7. A procedure for approximating sampling distributions (which can then be used to
construct confidence intervals) when theory cannot tell us their shape is
A) least squares.
B) the bootstrap.
C) residual analysis.
D) standardization.
8. A medical researcher is working on a new treatment for a certain type of cancer. The
average survival time after diagnosis on the standard treatment is two years. In an early trial, she tries
the new treatment on three subjects who have an average survival time after diagnosis of four years.
Although the survival time has doubled, the results are not statistically significant even at the 0.10
significance level. Suppose, in fact, that the new treatment does increase the mean survival time in the
population of all patients with this particular type of cancer. The researcher has
A) committed a type I error.
B) committed a type II error.
C) incorrectly used a level 0.10 test when she should have used a 0.05 level test.
D) incorrectly used a level 0.10 test when she should have computed the P-value.
Use the following to answer question 9:
A researcher wishes to determine if students are able to complete a certain pencil-and-paper maze more
quickly while listening to classical music. Suppose the time (in seconds) needed for high school
students to complete the maze while listening to classical music follows a normal distribution with
mean μ and standard deviation s = 4. Suppose also that in the general population of all high school
students the time needed to complete the maze (without listening to classical music) follows a normal
distribution with mean 40 and standard deviation s = 4. The researcher, therefore, decides to test the
hypotheses
H0: μ = 40, Ha: μ < 40
To do so, the researcher has 10,000 high school students complete the maze with classical music
playing. The mean time for these students is J = 39.8 seconds and the P-value is less than 0.0001.
9. Suppose that two high school students decide to see if they get the same results as the researcher.
They both take the maze while listening to classical music. The mean of their times is J = 39.8 seconds,
the same as that of the researcher. It is appropriate to conclude which of the following?
A) The students have reproduced the results of the researcher, and their P-value will be the
same as that of the researcher.
B) The students have reproduced the results of the researcher, but their P-value will be slightly
smaller than that of the researcher.
C) The students will reach the same statistical conclusion as the researcher, but their Pvalue
will be a bit different than that of the researcher.
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Test 10B
D) None of the above.
10. In a test if statistical hypotheses, the P-value tells us
A) if the null hypothesis is true.
B) if the alternative hypothesis is true.
C) the largest level of significance at which the null hypothesis can be rejected.
D) the smallest level of significance at which the null hypothesis can be rejected.
11. A certain population follows a normal distribution with mean μ and standard deviation s= 2.5. You
collect data and test the hypotheses
H0: μ = 1, Ha: μ ¹ 1
You obtain a P-value of 0.022. Which of the following is true?
A) A 95% confidence interval for μ will include the value 1.
B) A 95% confidence interval for μ will include the value 0.
C) A 99% confidence interval for μ will include the value 1.
D) A 99% confidence interval for μ will include the value 0.
12. An engineer designs an improved light bulb. The previous design had an average
lifetime of 1200 hours. The mean lifetime of a random sample of 2000 new bulbs is
found to have a mean lifetime of 1201 hours. Although the difference from the old mean
lifetime of 1200 hours is quite small, the P-value is 0.03 and the effect is statistically
significant at the 0.05 level. If, in fact, there is no difference between the mean lifetimes of the new and
old designs, the researcher has
A) committed a type I error.
B) committed a type II error.
C) a probability of being correct that is equal to the P-value.
D) a probability of being correct that is equal to 1 – (P-value).
Part 2: Free Response
Communicate your thinking clearly and completely.
9. A steel mill’s milling machine produces steel rods that are supposed to be 5 cm in diameter. When
the machine is in statistical control, the rod diameters vary according to a normal distribution with
mean µ = 5 cm and standard deviation  = 0.02 cm. A large sample of 150 produced by the
machine yields a sample mean diameter of 5.005 cm.
(a) Construct a 99% confidence interval for the true mean diameter of the rods produced by the
milling machine. Follow the inference toolbox.
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Test 10B
(b) Does the interval in (a) give you reason to suspect that the machine is not producing rods of the
correct diameter? State appropriate hypotheses and a significance level. Then explain your
conclusion.
(c) Describe a Type II error in the context of this problem. How could the manufacturer decrease
the probability of a Type II error.
10. A pharmaceutical manufacturer does a chemical analysis to check the potency of products. The
standard release potency for cephalothin crystals is 910. An assay of 16 lots gives the following
potency data:
897
914
913
906
916
918
905
921
918
906
895
893
908
906
907
901
Assume a population standard deviation  = 8.2.
(a) Construct a 99% confidence interval for the population mean. Follow the Inference Toolbox.
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Test 10B
You want to test hypotheses about the mean population potency,
H 0 : µ = 910
H a : µ < 910
(
)(
at the 1% level of significance. The z test statistic is z = x − 910 8.2
(b) What is the rule for rejecting H 0 in terms of z?
)
16 .
(c) What values of x would lead you to reject H 0 ?
(d) Describe a Type I error in the context of this problem. What is the probability of a Type I error?
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Test 10B