Download A study was conducted using data collected on the birth weights of a

Review Ch 11-12 MC
AP STATS
1.
Name___________________
Date________Per_____
A medical society conducts a study to determine if heart attack sufferers who arrive at the city hospital on nights and weekends wait longer
for an artery-clearing angioplasty than patients who arrive at the city hospital during regular hours. Current guidelines recommend that
patients wait no longer than 90 minutes from the time they enter the emergency room. They hypotheses for this study were
H o : 1  2
H o : 1  2 , where 1  true mean waiting time for heart attack sufferers who would arrive at the city
hospital at night or on weekends and  2  true mean waiting time for heart attack sufferers who would arrive at the city hospital during
and
regular hours. If was found that the mean waiting time for the random sample of those going to the hospital at night or on the weekend
was
x1  116 minutes, while those in the random sample arriving during regular hours had a mean waiting time of x2  95
minutes.
The p-value for the resulting test was 0.016. Which one of the following is a correct conclusion of this test?
2.
x1  x2
1  2 .
A
Since
B
There is enough evidence at the 5% level to conclude that
C
The observed difference between
D
Since the p-value is 0.016, we would fail to detect any difference between the two means.
E
Only 1.6% of the time will
we can conclude that
x1
and
x2
is significant at
  0.01.
1  2 .
Do children diagnosed with attention deficit/hyperactivity disorder have smaller brain volumes than children with out the condition? The
following data come from research described in the article “Developmental Trajectories of Brain Volume Abnormalities in Children and
Adolescents with ADHD.” Brain scans were completed on 152 children with ADHD and 139 children of similar age without ADHD. Do
these data provide convincing evidence that the mean brain volume of children with ADHD is smaller than that of children without ADHD?
Let
Let
1  the true mean brain volume of children with ADHD
 2  the true mean brain volume of children without ADHD
The correct Null and Alternate hypothesis is:
A
H :   0 and H :   0
0
3.
1  2 ,
D
A
D
C
H 0 : 1  2 and H A : 1  2
E
H 0 : 1  2  D and H A : 1  2  D
B
H 0 : 1  2 and H A : 1  2
D
H 0 : 1  2 and H A : 1  2
In a random sample of 60 shoppers chosen from the shoppers at a large suburban mall, 36 indicated that they had been to a movie in the past
month. In an independent random sample of 50 shoppers chosen from the shoppers in a large downtown shopping area, 31 indicated that
they had been to a movie in the past month. What significance test should be used to determine whether these data provide sufficient
evidence to reject the hypothesis that the proportion of shoppers at the suburban mall who had been to a movie in the past month is the same
as the proportion of shoppers in a large downtown shopping area who had been to a movie in the past month?
A
one-proportion z interval
B two-proportion z interval
C
two-sample t test
D one-proportion z test
E
two-proportion z test
4.
5.
6.
7.
A study done by the Duke University Medical School examined the effectiveness of St. John’s Wort as a natural alternative to reducing
depression. In a double-blind study, 100 people with mild to moderate depression were given daily doses of St. John’s Wort. Another 100
people with similar symptoms were given a sugar pill. Assignments to the two treatment groups were made randomly. Each participant was
asked to count the number of days they exhibited feelings of depression. A two-sample t test on the difference in the mean number of days
of depression was performed. The P-value of 0.42. Which of the following is a correct interpretation of the P-value?
A
About 42% of the time, samples drawn from population with no differences in mean depression count would show a difference at least
as extreme as that found in the study.
B
Forty-two percent of the participants did not experience any relief.
C
There was a 42% drop in the mean number of days of depression in the control group.
D
There was a 42% drop in the mean number of days of depression in the group that received the St. John’s Wort.
E
There was a 42% drop in the difference in the number of days of depression between the two groups.
The dentists in a dental clinic would like to determine if there is a difference between the number of new cavities in people who eat an
apple a day and in people who eat less than one apple a week. They are going to conduct a study with 50 people in each group. Fifty clinic
patients who report that they routinely eat an apple a day and 50 clinic patients who report that they eat less than one apple a week will be
identified. The dentists examine the patients and their records to determine the number of new cavities the patients have had over the past
two years. What significance test should be used to determine if there is a difference between the number of new cavities in people who eat
an apple a day and in people who eat less than one apple a week?
A
One-sample t-test
B Two-proportion z-test
D
Two-sample t-test
E Paired t-test
A policeman believes that more than 40% of older drivers speed on highways, but a confidential survey found that 49 of 88 randomly
selected older drivers admitted to speeding on highways at least once. What significance test should be used to determine whether the
data provides sufficient evidence to reject the null hypothesis that the proportion of older drivers who speed on highways is 40%?
A
One-sample t-test
B Two-proportion z-test
D
One-proportion z-test
E Paired t-test
C Chi-squared test for homogeneity
Sophomore, junior, and senior students at a high school will be surveyed regarding a potential increase in the extracurricular student
activities fee. There are three possible responses to the survey question---agree with the increase, do not agree with the increase, no opinion.
A chi-square test will be conducted to determine whether the response to this question is independent of the class in which the student is a
member. How many degrees of freedom should the chi-square test have?
A
8.
C Chi-squared test for homogeneity
9
B 6
C 4
D 2
E 1
Each person in a random sample of adults indicated his or her favorite color. The results, shown in the table below, are reported by age
group of the respondents.
If choice of color is independent of age group, which of the following expressions is equal to the expected number of respondents who are
aged 30 to 50, inclusive, and prefer green?
A
 99 108
314
B
 69 108
314
C
 35 99 
108
D
 35 108 
314
E
 35 99 
314
9.
Many times the weather person on the local news station weather report will give several different high temperatures (in degrees Fahrenheit)
for the same city, such as the high temperature downtown and the high temperature at the airport. Let
temperature downtown and
2
1
=the true mean daily high
=the true mean daily high temperature at the airport. Based on a random sample of 200 days from the past
six years, a 95% confidence interval for the mean differences,
constructed. This interval is given by
d  2  1 , between the airport and downtown temperatures was
 5.3, 1.9 . Which of the following is a correct conclusion based on this interval?
A
The mean temperature at the airport is only 95% as warm as the mean temperature downtown.
B
The mean temperature at the airport is cooler than the temperature downtown on 95 out of every 100 days.
C
Ninety-five percent of the time, the true mean difference in temperatures is between
D
We are 95% confident that the mean temperature at the airport is between
downtown.
1.9
and
5.7 cooler than the mean temperature
E
We are 95% confident that the mean temperature at the airport is between
downtown.
1.9
and
5.7 warmer than the mean temperature
1.9
and
5.7 .
Use the following to answer questions 10 through 12:
Some researchers have conjectured that stem-pitting disease in peach tree seedlings might be controlled with weed and soil treatment. An experiment
was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In a field containing 20 seedlings, 10
were randomly selected from throughout the field and assigned to receive Herbicide A. The remaining 10 seedlings were to receive Herbicide B. Soil
and weeds for each seedling were treated with the appropriate herbicide, and at the end of the study period, the height (in centimeters) was recorded
for each seedling. A box plot of each data set showed no indication of non-normality. The following results were obtained:
Herbicide A:
Herbicide B:
X 1 = 94.5 cm
X 2 = 109.1 cm
s1 = 10 cm
s2 = 9 cm
10. Referring to the information above, a 95% confidence interval for 2 – 1 (using the conservative value for the
degrees of freedom) is
A) 14.6 ± 7.36. B) 14.6 ± 7.80. C) 14.6 ± 9.62. D) 14.6 ± 13.93.
E) 14.6 ± 33.18.
11. Referring to the information above, suppose we wished to determine if there tended to be a significant difference in
mean height for the seedlings treated with the different herbicides. To answer this question, we decide to test the
hypotheses H0: µ2 – µ1 = 0, Ha: µ2 – µ1 ≠ 0. Based on our data, the value of the two-sample t test statistic is
A) 14.60. B) 7.80. C) 3.43. D) 2.54. E) 1.14.
12. Referring to the information above, suppose we wished to determine if there tended to be a significant difference in
mean height for the seedlings treated with the different herbicides. To answer this question, we decide to test the
hypotheses H0: µ2 – µ1 = 0, Ha: µ2 – µ1 ≠ 0. The 90% confidence interval is 14.6 ± 7.80 cm. Based on this confidence
interval,
A) we would not reject the null hypothesis of no difference at the 0.10 level.
B) we would reject the null hypothesis of no difference at the 0.10 level.
C) we would reject the null hypothesis of no difference at the 0.05 level.
D) the P-value is less than 0.10.
E) both C) and D) are correct.
13.
Based on a random sample of 50 students, the 90 percent confidence interval for the mean amount of money students spend on lunch at a
certain high school is found to be ($3.45, $4.15). Which of the following statements is true?
A
90% of the time, the mean amount of money that all students spend on lunch at this high school will be between $3.45 and $4.15.
B
90% of all students spend between $3.45 and $4.15 on lunch at this high school.
C
90% of all random samples of 50 students obtained at this high school would result in a sample mean amount of money students spend
on lunch between $3.45 and $4.15.
D
90% of all random samples of 50 students obtained at this high school would result in a 90% confidence interval that contains the true
mean amount of money students spend on lunch.
E
Approximately 45 of the 50 students in the random sample will spend between $3.45 and $4.15 on lunch at this high school.
Use the following to answer questions 14 through 17:
A researcher wished to test the effect of the addition of extra calcium to yogurt on the “tastiness” of yogurt. A collection of 200 adult volunteers was
randomly divided into two groups of 100 subjects each. Group 1 tasted yogurt containing the extra calcium. Group 2 tasted yogurt from the same
batch as group 1 but without the added calcium. Both groups rated the flavor on a scale of 1 to 10, with 1 being “very unpleasant” and 10 being “very
pleasant.” The mean rating for group 1 was X 1 = 6.5 with a standard deviation of s1 = 1.5. The mean rating for group 2 was X 2 = 7.0 with a
standard deviation of s2 = 2.0. Assume the two groups’ ratings are independent. Let µ1 and µ2 represent the mean ratings we would observe for the
entire population represented by the volunteers if all members of this population tasted, respectively, the yogurt with and without the added calcium.
14. Referring to the information above, assuming two sample t procedures are safe to use, a 90% confidence interval for
µ1 – µ2 (using the conservative value for the degrees of freedom) is
A) –0.5 ± 0.25. B) –0.5 ± 0.32. C) –0.5 ± 0.42. D) –0.5 ± 0.5.
E) –0.5 ± 0.58.
15. Referring to the information above, suppose the researcher had wished to test the hypotheses H0: µ1 = µ2, Ha: µ1 < µ2.
The P-value for the test (using the conservative value for the degrees of freedom) is
A) larger than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
16. Referring to the information above, which of the following would lead us to believe that the t procedures were not
safe to use here?
A) The sample medians and means for the two groups were slightly different.
B) The distributions of the data were moderately skewed.
C) The data are integers between 1 and 10 and so cannot be normal.
D) Only the most severe departures from normality would lead us to believe the t procedures were not safe to use.
E) The standard deviations from both samples were very different from each other.
17. Referring to the information above, if we had used the more accurate software approximation to the degrees of
freedom, we would have used which of the following as the number of degrees of freedom for the t procedures?
A) 199. B) 198. C) 190. D) 183. E) 99.
Use the following to answer questions 18 through 20:
A sports writer wished to see if a football filled with helium travels farther, on average, than a football filled with air. To test this, the writer used 18
adult male volunteers. These volunteers were randomly divided into two groups of nine subjects each. Group 1 kicked a football filled with helium to
the recommended pressure. Group 2 kicked a football filled with air to the recommended pressure. The mean yardage for group 1 was
X
1
= 30
yards with a standard deviation of s1 = 8 yards. The mean yardage for group 2 was X 2 = 26 yards with a standard deviation of s2 = 6 yards. Assume
the two groups of kicks are independent. Let µ1 and µ2 represent the mean yardage we would observe for the entire population represented by the
volunteers if all members of this population kicked, respectively, a helium- and an air-filled football.
18. Referring to the information above, assuming two sample t procedures are safe to use, a 99% confidence interval for
µ1 – µ2 (using the conservative value for the degrees of freedom) is
A) (-.7, .7) yards. B) (-2.2, 10.2) yards. C) (-3.7, 11.7) yards. D) (-4.6, 12.6) yards.
E) (-5.84, 13.84) yards.
19. Referring to the information above, suppose the researcher had wished to test the hypotheses H0: µ1 = µ2, Ha: µ1 > µ2.
The P-value for the test (using the conservative value for the degrees of freedom) is
A) larger than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
20. Referring to the information above, to which of the following would it have been most important that the subjects be
blind during the experiment?
A) The identity of the sports writer.
B) Whether or not the balls were of regulation size and weight.
C) The method they were to use in kicking the ball.
D) Whether the ball they were kicking was filled with helium or air.
E) The direction in which they were to kick the ball.
21. A researcher wished to compare the effect of two stepping heights (low and high) on heart rate in a step-aerobics
workout. A collection of 50 adult volunteers was randomly divided into two groups of 25 subjects each. Group 1 did a
standard step-aerobics workout at the low height. The mean heart rate at the end of the workout for the subjects in
group 1 was
X
1
= 90.00 beats per minute with a standard deviation of s1 = 9 beats per minute. Group 2 did the same
workout but at the high step height. The mean heart rate at the end of the workout for the subjects in group 2 was X
2 = 95.08 beats per minute with a standard deviation of s2 = 12 beats per minute. Assume the two groups are
independent and the data are approximately normal. Let µ1 and µ2 represent the mean heart rates we would observe
for the entire population represented by the volunteers if all members of this population did the workout using the low
or high step height, respectively. Suppose the researcher had wished to test the hypotheses H0: µ1 = µ2, Ha: µ1 < µ2.
The P-value for the test (using the conservative value for the degrees of freedom) is
A) larger than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) less than 0.001.
C) between 0.01 and 0.05.
22. An SRS of 100 of a certain popular model car in 1993 found that 20 had a certain minor defect in the brakes. An SRS
of 400 of this model car in 1994 found that 50 had the minor defect in the brakes. Let p1 and p2 be the proportion of
all cars of this model in 1993 and 1994, respectively, that actually contain the defect. A 90% confidence interval for
p1 – p2 is 0.075 ± 0.071.
Suppose the sample of 1993 cars consisted of only 10 cars, of which two had the minor brake defect. Suppose also the
sample of 1994 cars consisted of only 40 cars, of which five had the minor brake defect. A 90% confidence interval
for p1 – p2 is now
A) the same as that for the original sample of 100 and 400 cars.
B) much wider than that for the original sample of 100 and 400 cars.
C) the same as 99% for the original sample of 100 and 400 cars.
D) unsafe to compute, since it is unsafe to use the normal distribution to approximate the sampling distribution of
p̂1 – p̂2 .
E)
much narrower than that for the original sample of 100 and 400 cars.
Use the following to answer questions 23 and 24:
An SRS of 100 flights by Airline 1 showed that 64 were on time. An SRS of 100 flights by Airline 2 showed that 80 were on time. Let p1 be the
proportion of on-time flights for all Airline 1 flights, and let p2 be the proportion of all on-time flights for all Airline 2 flights.
23. Referring to the information above, a 95% confidence interval for the difference p1 – p2 is
A) –0.16 ± 0.062. B) –0.16 ± 0.122. C) –0.16 ± 0.104. D) –0.16 ± 0.103.
E) 0.16 ± 0.062.
24. Referring to the information above, is there evidence of a difference in the on-time rate for the two airlines? To
determine this, you test the hypotheses H0: p1 = p2, Ha: p1 ≠ p2.
The P-value of your test is
A) greater than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
Use the following to answer questions 25 and 26:
A sociologist is studying the effect of having children within the first three years of marriage on the divorce rate. From city marriage records, she
selects a random sample of 400 couples that were married between 1985 and 1990 for the first time, with both members of the couple being between
the ages of 20 and 25. Of the 400 couples, 220 had at least one child within the first three years of marriage. Of the couples that had children, 83 were
divorced within five years, while of the couples that didn’t have children, only 52 were divorced within three years. Suppose p1 is the proportion of
couples married in this time frame that had a child within the first three years and were divorced within five years and p2 is the proportion of couples
married in this time frame that did not have a child within the first two years and were divorced within five years.
25. Referring to the information above, the estimate of p1 – p2 is
A) 0.0775. B) 0.0884. C) 0.3100. D) 0.3375. E) 0.3773.
26. Referring to the information above, the sociologist had hypothesized that having children early would increase the
divorce rate. She tested the one-sided alternative and obtained a P-value of 0.0314. The correct conclusion is that
A) if you want to decrease your chances of getting divorced, it is best to wait several years before having children.
B) having more children increases the risk of divorce during the first 5 years of marriage.
C) if you want to decrease your chances of getting divorced, it is best not to marry when you are closer to 30 years
old.
D) it is best not to have children.
E) there is evidence of an association between divorce rate and having children early in a marriage.
Use the following to answer questions 27 through 31:
I teach a large introductory statistics course. In the past, the proportions of students that received grades of A, B, C, D, or F have been, respectively,
0.20, 0.30, 0.30, 0.10, and 0.10. This year, there were 200 students in the class, and I gave them the following grades.
Grade
Number
A
56
B
74
C
60
D
9
F
1
I wish to test to see whether the distribution of grades this year was different from the distribution in the past. To do so, I plan to use the χ2 statistic.
27. Assuming that the χ2 statistic has approximately a χ2 distribution, how many degrees of freedom does the distribution
have?
A) 200. B) 199. C) 9. D) 5. E) 4.
28. The component (O – E)2/E of the χ2 statistic corresponding to a grade of C is
A) 0.
B) 1.
C) 33.77. D) 30.
E) 11,880.30.
29. I compute the value of the χ2 statistic to be 33.77. The P-value of the test is
A) greater than 0.20.
D) between 0.01 and 0.05.
B) between 0.10 and 0.20.
E) less than 0.01.
C) between 0.05 and 0.10.
30. The grade category that contributes the largest component to the χ2 statistic is
A) A. B) B. C) C. D) D. E) F.
31. I may assume that the χ2 statistic has an approximate χ2 distribution because of which of the following?
A) The expected number of people in each grade category is greater than 5.
B) The sample size is 200, which is large enough for the approximation to be valid.
C) The number of categories is small relative to the number of observations.
D) I may not assume that the χ2 statistic has an approximate χ2 distribution, because there is only one person in the
F grade category.
E) The expected value of the χ2 statistic is greater than 10.
Use the following to answer questions 32 and 33 :
Using computer software, I generate 1000 random numbers that are supposed to follow a standard normal distribution. I classify these 1000 numbers
according to whether their values are less than 0 or greater than or equal to 0. The results are given in the table below.
Less Than 0
Greater Than or Equal to 0
Number
512
488
Because the standard normal distribution is symmetric about 0, I would expect half of the random numbers generated to be less than 0 and half to be
greater than or equal to 0. To test to see if the distribution of the observed number in each category differs significantly from the expected distribution
of counts, I use the χ2 statistic.
32. The value of the χ2 statistic is
A) 0.024. B) 0.048. C) 0.288.
D) 0.576.
E) 1.152.
33. In this case, the χ2 statistic has approximately a χ2 distribution. How many degrees of freedom does this distribution
have?
A) 0. B) 1. C) 2. D) 999. E) None of the above.
Use the following to answer questions 34 through 38:
Are avid readers more likely to wear glasses than those who read less frequently? Three hundred men in the Korean army were selected at random
and classified according to whether or not they wore glasses and whether the amount of reading they did was above average, average, or below
average. The results are presented in the following table.
Wear Glasses?
Amount of Reading
Yes
No
Above average
47
26
Average
48
78
Below average
31
70
34. This is an r × c table. The number r has value
A) 2. B) 3. C) 4. D) 6. E) 8.
35. The proportion of men in the table who wear glasses is
A) 0.24. B) 0.37. C) 0.38. D) 0.42. E) 0.64.
36. The proportion of all above-average readers who wear glasses is
A) 0.24. B) 0.27. C) 0.37. D) 0.42. E) 0.64.
37. Suppose we wish to test the null hypothesis that there is no association between the amount of reading you do and
whether or not you wear glasses. Under the null hypothesis, the expected number of above-average readers who wear
glasses is approximately
A) 81.1.
B) 47.
C) 30.7.
D) 27.2.
E) 19.7.
38. Suppose we wished to display in a graph the proportion of all above-average readers who wear glasses and do not
wear glasses, respectively. Which of the following graphical displays is best suited to this purpose?
A) A stemplot.
B) A scatterplot.
C) A standard normal plot.
D) A bar graph.
E) A boxplot.
39. Recent revenue shortfalls in a Midwestern state led to a reduction in the state budget for higher education. To offset
the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was
needed simply to compensate for the lost support from the state. Random samples of 50 freshmen, 50 sophomores, 50
juniors, and 50 seniors from the university were asked whether or not they were strongly opposed to the increase,
given that it was the minimum increase necessary to maintain the university’s budget at current levels. The results are
given in the following table.
Strongly Opposed?
Yes
No
Freshman
39
11
Year
Sophomore
36
14
Junior
29
21
Senior
18
32
Which hypotheses are being tested by the chi-square test?
A)
B)
C)
D)
E)
The null hypothesis is that the closer students get to graduation, the less likely they are to be opposed to tuition
increases. The alternate is that how close a student is to graduation makes no difference in the student’s
opinion.
The null hypothesis is that the mean number of students who are strongly opposed is the same for each of the
four years. The alternative is that the mean is different for at least two of the four years.
The null hypothesis is that the distribution of whether or not a student is strongly opposed is the same for each
of the four years. The alternative is that the distribution is different for at least two of the four years.
The null hypothesis is that the distribution of the total number of students sampled is the same for each of the
four years. The alternative is that the distribution is different for at least two of the four years.
The null hypothesis is that a student’s year in school and whether or not the student is strongly opposed are
independent. The alternative is that these variables are dependent.
40. A type II error is made by
a. Rejecting Ho when it is true.
b. Rejecting Ho when it is false.
c. Failing to reject Ho when it is true
d. Failing to reject Ho when it is false.
e. None of the above.
ANSWERS:
1. B
9. D
17. D
25. B
33. B
2. B
10. C
18. E
26. E
34. B
3. E
11. C
19. A
27. E
35. D
4. A
12. B
20. D
28. A
36. E
5. D
13. D
21. B
29. E
37. C
6. D
14. C
22. D
30. E
38. D
7. C
15. C
23. B
31. A
39. C
8. A
16. D
24. C
32. D
40. D