Name ________________________________________________________________ Period __________
AP Statistics
Unit 8 Review
Vocabulary:
test of significance statistically significant test statistic p‐value significance level null hypothesis alternative hypothesis Type I Error Type II Error Power 1.
What is a null hypothesis?
2.
What is an alternative hypothesis?
3.
In statistics, what is meant by the P-value?
4.
If a P-value is small, what do we conclude about the null hypothesis?
5.
If a P-value is large, what do we conclude about the null hypothesis?
6.
How small should the P-value be in order to claim that a result is statistically significant?
7.
Explain the difference between a one-sided alternative hypothesis and a two-sided alternative
hypothesis.
8.
What is meant by a significance level?
9.
What is the relationship between the significance level  and the probability of Type I Error?
10.
When would you use a 1-sample t-procedure?
11.
What is a matched-pairs t-procedure? How is it different than a two sample problem?
12.
In a two-sample problem, must/should the two sample sizes be equal?
13.
In a two-sample problem, what is the null hypothesis for comparing two means?
14.
In a matched pairs problem, what is the null hypothesis?
15.
For a one-sample hypothesis test where H 0 : p  p0 , what is the z test statistic?
16.
What assumptions must be met in order to use z procedures for inference about two proportions?
17.
Describe how to increase the power of a significance test.
18.
Explain the difference between a Type I Error and a Type II Error.
Multiple Choice:
1.
Which of the following is INCORRECT about the use of a paired experiment?
a.
The object of pairing (or blocking) is to account for the effect of possible other factors (such as
fertility of soils).
b.
The analysis of paired data starts by finding the difference between the values of the pair. The
order of the difference (as long as it is consistent) is unimportant.
c.
It is crucial to recognize pairing. If pairing is not recognized, the results will not be as accurate
and precise as possible.
d.
The degrees of freedom is equal to the number of pairs minus 1.
e.
Because pairing is beneficial, we can pair all data by matching the smallest value of each sample,
the second smallest value of each sample, the third smallest value of each sample, etc.
2.
The average time it takes for a person to experience pain relief from aspirin is 25 minutes. A new
ingredient is added to help speed up relief. Let  denote the average time to obtain pain relief with the
new product. An experiment is conducted to verify if the new product is better. What are the null and
alternative hypotheses?
H 0 :   25
H 0 :   25
H 0 :   25
a.
b.
c.
H a :   25
H a :   25
H a :   25
H 0 :   25
H 0 :   25
d.
e.
H a :   25
H a :   25
3.
We wish to test H 0 that the average family income of Manitoba families is at least $15,000 at level of
significance   0.05 . In order to test the null hypothesis a sample size of 1000 is selected from the
population, and the p-value of the test is determined to be 0.02. We then:
a.
reject H 0 because the data are sufficiently unusual if the null hypothesis were false.
b.
reject H 0 because the data are sufficiently unusual if the null hypothesis were true.
c.
fail to reject H 0 because the data are not sufficiently unusual if the null hypothesis were true.
d.
fail to reject H 0 because the data are not sufficiently unusual if the null hypothesis were false.
e.
reject H 0 because the data are sufficiently unusual.
4.
A group of nutritionists is hoping to prove that a new soy bean compound has more protein per gram
than roast beef, which has a mean protein content of 20. A random sample of 5 batches of the soy
compound have been tested, with the following results:
Protein content:
15 22 17 19 23 What assumption(s) do we have to make in order to carry out a legitimate statistical test of the
nutritionists’ claim?
a.
The observations are from a normally distributed population.
b.
The mean protein content of the 5 batches follows a normal distribution.
c.
The variance of the population is known.
d.
Both (a) and (b) must be assumed
d.
Both (a), (b), and (c) must be assumed
5.
An appropriate 95% confidence interval for  has been calculated as  0.73,1.92  based on n  15
observations from a population with a normal distribution. The hypothesis of interest are H 0 :   0
versus H a :   0 . Based on this confidence interval,
a.
we should reject H 0 at the   0.05 level of significance
b.
we should not reject H 0 at the   0.05 level of significance
c.
we should reject H 0 at the   0.10 level of significance
d.
we should not reject H 0 at the   0.10 level of significance
e.
we cannot perform the required test since we don’t know the value of the test statistic.
6.
A random sample of 100 votes in a community produced 59 voters in favor of candidate A. The
observed value of the test statistic for testing the null hypothesis H 0 : p  0.5 versus the alternative
hypothesis H a : p  0.5 is:
a. 1.80
b. 1.90
c. 1.83
d. 1.28
e. 1.75
7.
In a test of H 0 : p  0.4 against H a : p  0.4 , a sample of size 100 produces z  1.28 for the value of the
test statistic. Thus the p-value of the test is approximately equal to:
a. 0.90
b. 0.40
c. 0.05
d. 0.20
e. 0.10
8.
We wish to test if a new feed increases the mean weight gain compared to an old feed. At the
conclusion of the experiment it was found that the new feed gave a 10 kg bigger gain than the old feed.
A two-sample t-test with the proper one-sided alternative was done and the resulting p-value was 0.082.
This means:
a.
There is an 8.2% chance the null hypothesis is true.
b.
There was only an 8.2% chance of observing an increase greater than 10 kg (assuming the null
hypothesis was true.
c.
There was only an 8.2% chance of observing an increase greater than 10 kg (assuming the null
hypothesis was false).
d.
There is an 8.2% chance that alternate hypothesis is true.
e.
There is only an 8.2% chance of getting a 10 kg increase.
9.
Following the analysis of some data on two samples drawn from populations in which the variable of
interest is normally distributed, the p-value for the comparison of the two sample means under the null
hypothesis that the two population’s means are equal against the alternative hypothesis that the two
population means are not equal was found to be 0.0063. This p-value indicates that:
a.
there is very little evidence in the data for a conclusion to be reached.
b.
there is rather strong evidence against the null hypothesis.
c.
the evidence against the null hypothesis is not strong.
d.
the null hypothesis should be accepted.
e.
there is rather strong evidence against the alternative hypothesis.
10.
You want to design a study to estimate the proportion of students on your campus who agree with the
statement, “The student government is an effective organization for expressing the needs of students to
the administration.” You will use a 95% confidence interval and you would like the margin of error of
the interval to be 0.05 or less. The minimum sample size required is approximately
a. 22
b. 1795
c. 385
d. 271
11.
A significance test was performed to test the null hypothesis H 0: : p  0.5 versus the alternative
H a : p  0.5 . The test statistic is z = 1.40. Which of the following is closest to the P-value for this test?
a) 0.0808
b) 0.1492
c) 0.1616
d) 0.2984
e) 0.9192
12.
A test of H0: μ = 60 versus Ha: μ ≠ 60 produces a sample mean of x = 58 and a P-value of 0.04. At an
  0.05 level, which of the following is an appropriate conclusion?
a)
There is sufficient evidence to conclude that μ < 60.
b)
There is sufficient evidence to conclude that μ = 60.
c)
There is insufficient evidence to conclude that μ = 60.
d)
There is insufficient evidence to conclude that μ ≠ 60.
e)
There is sufficient evidence to conclude that μ ≠ 60.
13.
Because t procedures are robust, the most important condition for their use is
a)
the population standard deviation is known.
b)
the population distribution is approximately Normal.
c)
the data can be regarded as a random sample from the population.
d)
np and n(1 – p) are both at least 10.
e)
all values in the sample are within two standard deviations of the mean.
14.
We want to test H0: μ = 1.5 vs. Ha : μ ≠ 1.5 at α = 0.05. A 95% confidence interval for μ calculated from
a given random sample is (1.4, 3.6). Based on this finding we
a)
fail to reject H0 .
b)
reject H0 .
c)
cannot make any decision at all because the value of the test statistic is not available.
d)
cannot make any decision at all because the distribution of the population is unknown.
e)
cannot make any decision at all because (1.4, 3.6) is only a 95% confidence interval for μ.
15.
Which of the following statements is/are correct?
I. The power of a significance test depends on the effect size.
II. The probability of a Type II error is equal to the significance level of the test.
III. Error probabilities can be expressed only when a significance level has been specified.
a)
I and II only
b)
I and III only
c)
II and III only
d)
I, II, and III
e)
None of the above gives the complete set of correct responses.
Use the following for questions 16 and 17:
The water diet requires you to drink two cups of water every half hour from the time you get up until you go to
bed, but otherwise allows you to eat whatever you like. Four adult volunteers agree to test the diet. They are
weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are
Subject
Weight before diet
Weight after 6 weeks
A
180
170
B
125
130
C
240
125
D
150
152
16.
Which of the following conditions must be met in order to use a t-procedure on these paired data?
a)
Only the distribution of pre-diet weights must be approximately Normal.
b)
Only the distribution of differences (after 6 weeks – before) must be approximately Normal.
c)
The distribution of both pre-diet weights and six-week weights must be approximately Normal.
d)
The distribution of pre-diet weights and the distribution of differences (after 6 weeks – before)
must be approximately Normal.
e)
All three distributions—before diet, after 6 weeks, and the difference—must be approximately
Normal.
17.
What would a Type II error be for this test of the water diet?
a)
Concluding that the diet leads to weight loss when it doesn’t.
b)
Concluding that the diet leads to weight loss when it really does.
c)
Not concluding that the diet leads to weight loss when it does.
d)
Not concluding that the diet leads to weight loss when it really doesn’t.
e)
Drawing a conclusion from this test when the Normality condition has not been satisfied.
18.
A researcher wishes to determine if people are able to complete a certain pencil and paper maze more
quickly while listening to classical music. Suppose previous research has established that the mean time
needed for people to complete a certain maze (without music) is 40 seconds. The researcher, therefore,
decides to test the hypotheses H0: μ = 40 versus Ha: μ < 40, where μ is the time in seconds needed to
complete the maze while listening to classical music. To do so, the researcher has 10,000 people
complete the maze with classical music playing. The mean time for these people is x = 39.92 seconds,
and the P-value of his significance test is 0.0002. Which statement below best describes the appropriate
conclusion to draw from this study?
a)
The researcher has proved that listening to classical music substantially improves the time it
takes to complete the maze.
b)
The researcher has strong evidence that listening to classical music substantially improves the
time it takes to complete the maze.
c)
The researcher has moderate evidence that listening to classical music substantially improves the
time it takes to complete the maze.
d)
Although the researcher has obtained a statistically significant result, it appears to have little
practical significance.
e)
Since the P-value is greater than the reciprocal of the sample size, this is not a significant result.
19.
The recommended daily Calcium intake for women over 21 (and under 50) is 1000 mg per day. The
health services at a college are concerned that women at the college get less Calcium than that, so they
take a random sample of female students in order to test the hypotheses H 0 :   1000 versus
H a :   1000 . Prior to the study they estimate that the power of their test against the alternative
H a :   900 is 0.85. Which of the following is the best interpretation of this value?
a)
The probability of making a Type II error.
b)
The probability of rejecting the null hypothesis when the parameter value is 1000.
c)
The probability of rejecting the null hypothesis when the parameter value is 900.
d)
The probability of failing to reject the null hypothesis when the parameter value is 1000.
e)
The probability of failing to reject the null hypothesis when the parameter value is 900.
20.
The following are percents of fat found in 5 samples of each of two brands of ice cream:
A
5.7
4.5
6.2
6.3
7.3
B
6.3
5.7
5.9
6.4
5.1
Which of the following procedures is appropriate to test the hypothesis of equal average fat content in
the two types of ice cream?
a) Paired t test with 5 df.
b) Two-sample t test with 4 df.
c) Paired t test with 4 df.
d) Two-sample t test with 9 df.
e) Two-proportion z test
21.
All of us nonsmokers can rejoice—the mosaic tobacco virus that affects and injures tobacco plants is
spreading! Meanwhile, a tobacco company is investigating if a new treatment is effective in reducing the
damage caused by the virus. Eleven plants were randomly chosen. On each plant, one leaf was randomly
selected, and one half of the leaf (randomly chosen) was coated with the treatment, while the other half
was left untouched (control). After two weeks, the amount of damage to each half of the leaf was
assessed. For purposes of comparing the damage, which of the following is the appropriate type of
procedure?
a)
Two-proportion z procedures
b)
Two-sample z procedures
c)
Matched pairs t procedures
d)
Two proportion t procedures
e)
Two-sample t procedures
22.
Phoebe has a theory that older students at her high school are more likely to bring a bag lunch than
younger students, because they have grown tired of cafeteria food. She takes a simple random sample of
80 sophomores and finds that 52 of them bring a bag lunch. A simple random sample of 104 seniors
reveals that 78 of them bring a bag lunch. Letting p1 be the proportion of sophomores who bring a bag
lunch, and p2 be the proportion of seniors who bring a bag lunch, Phoebe tests the hypotheses
H 0 : p1  p2  0 ; H a : p1  p2  0 at the α = 0.05 level. Phoebe’s test statistic is –1.48. Which of the
following is closest to the appropriate P-value for the test?
a) 0.0694
b) 0.0808
c) 0.1388
d) 0.8612
e) 0.9306
Free Response:
1.
Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal
means that the paper will live forever in libraries. The British Medical Journal combines the two: it
prints short and readable versions, with longer versions available online. Is this OK with authors? The
journal asked a random sample of 104 of its recent authors several questions. One question was “Should
the journal continue using this system?” In the sample, 72 said “Yes.”
a)
Do the data give good evidence that more than two-thirds (67%) of authors support continuing
this system? Carry out an appropriate test to help answer this question.
b)
Interpret the P-value from your test in the context of the problem.
2.
“Red tide” is a bloom of poison-producing algae—a few different species of a class of plankton called
dinoflagellates. When weather and water condition cause these blooms, shellfish such as clams living in
the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of
Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along
the coastline. If the mean level of toxin in clams exceeds 800μg (micrograms) of toxin per kg of clam
meat in any area at a 5% level of significance, clam harvesting is banned there until the bloom is over
and levels of toxin in clams subside. During a bloom, the distribution of toxin levels in clams on a single
mudflat is distinctly non-Normal.
a)
Define the parameter of interest and state appropriate hypotheses for the DMF to test.
b)
Because of budget constraints and the large number of coastal areas that must be tested, the DMF
would like to sample no more than 10 clams from any single area. Explain why this sample size
may lead to problems in carrying out the significance test from (a).
c)
Describe a Type I and a Type II error in this situation and the consequences of each.
d)
The DMF is considering changing the significance level of the test to 10%. Discuss the impact
this might have on error probabilities and the power of the test, and describe the practical
consequences of this change.
3.
The Environmental Protection Agency has determined that safe drinking water should contain no more
than 1.3 mg/liter of copper. You are testing water from a new source, and take 30 water samples. The
mean copper content in your samples is 1.36 mg/l and the standard deviation is 0.18 mg/l. There do not
appear to be any outliers in your data.
a)
Do these samples provide convincing evidence at the α = 0.05 level that the water from this
source contains unsafe levels of copper? Justify your answer.
b)
How would your conclusion change if your sample mean had been 1.355 mg/l? What point does
this make about statistical significance?
4.
A consumer advocacy group tests the mean vitamin C content of 50 different brands of bottled juices
using, in each case, a t-test of significance in which the null hypothesis is the mean amount of vitamin C
that is on the nutrition facts label for that brand of juice. They find that two of the 50 juice brands have
statistically significantly lower Vitamin C than claimed at the α = 0.05 level. Is this an important
discovery? Explain.
5.
Tai Chi is often recommended as a way to improve balance and flexibility in the elderly. Below are
before-and-after flexibility ratings (on a 1 to 10 scale, 10 being most flexible) for 8 men in their 80’s
who took Tai Chi lessons for six months.
Subject
Flexibility rating after Tai Chi
Flexibility rating before Tai Chi
A
2
1
B
4
2
C
3
1
D
3
2
E
3
1
F
4
4
G
5
2
H
10
6
Do these paired data adequately meet the Normality condition for a t-procedure? Justify your answer.
6.
A pharmaceutical company is testing a new drug for reducing cholesterol levels. To approve the drug for
the next round of testing, they need to show that this drug reduces mean total cholesterol level by at least
50 mg/dL. They initially plan a study that involves 50 subjects and a significance level of α = 0.05, but
they discover that the power of the test against this effect size is only 0.24. What are two ways they can
increase the power of the test without changing effect size?
7.
The pesticide diazinon is in common use to treat infestations of the German cockroach, Blattella
germanica. A study investigated the persistence of this pesticide on various types of surfaces.
Researchers applied a 0.5% emulsion of diazinon to glass and plasterboard. After 14 days, they
randomly assigned 72 cockroaches to two groups of 36, placed one group on each surface, and recorded
the number that died within 48 hours. On glass, 18 cockroaches died, while on plasterboard, 25 died.
a)
Construct and interpret a 90% confidence interval for the difference in the proportion of
cockroaches that die on each surface.
b)
Based only on this interval, do you think that the difference in proportion of cockroaches that
died on each surface is significant? Justify your answer.
8.
A study of iron deficiency among infants compared blood hemoglobin levels of a random sample of
one-year-old infants who had been breast-fed to a random sample of one-year old infants who had been
fed with standard infant formula. Here are the results.
Breast-fed infants
Formula-fed infants
n
23
19
x
13.3
12.4
s
1.7
1.8
a)
We wish to test the hypothesis H 0 :  B   F  0 against: H a :  B   F  0 , where  B and  F are the
population mean blood hemoglobin levels for breast-fed and formula-fed infants, respectively.
What additional information would you need to confirm that the conditions for this test have been met?
b)
Assuming the conditions have been met, calculate the test statistic and P-value for this test.
c)
Interpret the P-value in the context of this study, and draw the appropriate conclusion at the α = 0.05
level.
d)
Given your conclusion in part (c), which type of error, Type I or Type II, is it possible to make?
Describe that error in the context of this study.
Selected Answers
Multiple Choice
1. E
5. B
9. B
13. C
17. C
21. C
Free Response
State:
1a.
Plan:
Do:
Conclude:
1b.
2. B
6. A
10. C
14. A
18. D
22. A
3. B
7. D
11. A
15. B
19. C
4. B
8. B
12. E
16. B
20. B
H 0 : p  0.67 ; H a : p  0.67
1 proportion z test with   0.05
p  value  0.3143 ; z  0.4838
fail to reject H0
2a. H 0 :   800  g ; H a :   800  g
2b.
2c.
2d.
State: H 0 :   1.3 mg / l ; H 0 :   1.3 mg / l
Plan: 1 sample t-test with   0.05
Do: p  value  0.039 ; t  1.8257 ; df  30  1  29
Conclude: reject H0
3b. p  value  0.052 ; fail to reject H0
3a.
4. Not important since P  Type I error   0.05
5.
fairly symmetric with
no outliers
So the Normality condition is met
6. increase sample size, increase significance level
7a.
State: Estimate pP  pG
Plan: 2 Sample z interval with 90% confidence level
Do:  0.00807,0.38082 
Conclude:
7b. Since 0 is not in the interval, reject H0
8a. Need to know about skewness and outliers.
8b. p  value  0.1065 ; t  1.6537 ; df  37.5976
8c. Conclude:fail to reject H0 (You still need to interpret p-value)
8d. Type II error
a fairly linear Normal
probability plot