Midterm Review
IB Statistics
Use the following to answer questions 1 and 2:
A psychologist studied the number of puzzles subjects were able to solve in a five-minute period
while listening to soothing music. Let X be the number of puzzles completed successfully by a
subject. The psychologist found that X had the following probability distribution:
Value of X
Probability
1
0.1
2
0.6
3
0.2
4
0.1
1. Referring to the information above, the probability that a randomly chosen subject
completes at least 2 puzzles in the five-minute period while listening to soothing music
A) is 0.1 B) is 0.4 C) is 0.7 D) is 0.9 E) is impossible to compute
2. Referring to the information above, P(X < 2) has value
A) 0.1 B) 0.4 C) 0.6 D) 0.9 E) 1
Use the following to answer questions 3 through 5:
Let the random variable X be a randomly generated number with the uniform probability density
curve given below.
3. Referring to the information above, P(X = 0.65) is
A) 0 B) 0.025 C) 0.25 D) 0.65 E) 1
4. Referring to the information above, P(X  0.55) has value
A) 0 B) 0.45 C) 0.55 D) 1
E) The value cannot be determined
5. Referring to the information above, P(0.6 < X < 1.3) has value
A) 0.3 B) 0.4 C) 0.7 D) 0.8 E) 1.3
6. If X is binomial with parameters n = 20 and p = 1/5, the mean X of X is
A) 5 B) 4 C) 1.414 D) 1.732 E) 1.789
7. If X is binomial with parameters n = 20 and p = 1/5, the standard deviation X of X is
A) 5 B) 4 C) 1.414 D) 1.732 E) 1.789
8. In a certain game of chance, your chances of winning are 0.2. If you play the game five
times and outcomes are independent, the probability that you win at most once is about
A) 0.0819 B) 0.2 C) 0.3277 D) 0.7373 E) 0.8871
9. In a certain game of chance, your chances of winning are 0.2. If you play the game five
times and outcomes are independent, the probability that you win more than 3 times is
A) 0.993 B) 0.00672 C) 0.0032 D) 0.0004 E) none of these
10. In a certain game of chance, your chances of winning are 0.2. You play the game five
times and outcomes are independent. Suppose it costs $1 to play the game each time.
Each time you win, you receive $4 (for a net gain of $3). Each time you lose, you
receive nothing (for a net loss of $1). Your expected winnings for five plays are
A) $1 B) $0 C) –$1 D) –$2 E) –$3
Use the following to answer questions 11 through 18:
A survey asks a random sample of 2000 adults in Mississippi if they support an increase in the
state sales tax from 3.5% to 4.5%, with the additional revenue going to education. For #11-15, let
X denote the number in the sample that say they support the increase. For #16-18, let p̂ denote
the proportion in the sample that say they support the increase. Suppose that 30% of all adults
in Mississippi support the increase.
11. The mean of X is
A) 3.5% B) 4.5%
C) 20.49
D) 600
E) 1400
12. The standard deviation of X is about
A) 3.5% B) 4.5% C) 20.49 D) 600
E) 1400
13. The probability that X is exactly 600 is about
A) 0 B) 0.0075 C) 0.0195 D) 0.511 E) 0.678
14. The probability that X is at most 600 is about
A) 0.0072 B) 0.3372 C) 0.4 D) 0.511
E) 0.9928
15. The probability that X is more than 650 is about
A) 0.0072 B) 0.3372 C) 0.4 D) 0.511 E) 0.9928
16. The mean  p̂ of p̂ is
A) 70% B) 30% ± 3.5%
C) 30% ± 4.5%
17. The standard deviation  p̂ of p̂ is about
A) 0.4 B) 0.24 C) 0.0102 D) 0.00033
D) 0.3
E) 600
E) 0.00016
18. The probability that p̂ is more than 0.50 is (Hint: find this probability by standardizing
0.50 and using a Normal distribution)
A) less than 0.0001 B) about 0.1 C) 0.1586 D) 0.4602 E) 0.5
19. A fair coin (one for which both the probability of heads and the probability of tails are
0.5) is tossed 200 times. Using binomcdf, the probability that less than 1/2 of the tosses
are heads is approximately
A) 0.9957 B) 0.5282 C) 0.4718 D) 0.09 E) 0.0067
20. Suppose we select an SRS of size n = 100 from a large population having proportion p
of successes. Let X be the number of successes in the sample. For which value of p
would it be safe to assume the sampling distribution of X is approximately normal?
A) 0.001 B) 0.09 C) 0.05 D) 0.01 E) 1/9
21. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. An
experimenter looks at each of 100 cards in turn, and the subject tries to read the
experimenter’s mind and name the shape on each card. What is the approximate
probability that the subject gets more than 28 correct if the subject does not have ESP
and is just guessing?
A) 0.7925 B) 0.2075 C) 0.104 D) 0.043 E) less than 0.0001
22. A multiple-choice exam has 100 questions, each with five possible answers. If a student
is just guessing at all the answers, the probability that he or she will get more than 25
correct is about
A) 0.2500 B) 0.1230 C) 0.0875 D) 0.0601 E) 0.0004
23. A college basketball player makes 85% of his free throws. Over the course of the season
he will attempt 100 free throws. Assuming free throw attempts are independent, the
probability that the number of free throws he makes exceeds 80 is approximately
A) 0.2000 B) 0.2266 C) 0.5 D) 0.7734 E) 0.8935
24. A college basketball player makes 80% of his free throws. Over the course of the season
he will attempt 300 free throws. Assuming free throw attempts are independent, what is
the approximate probability that he makes at least 251 of these attempts?
A) 0.99 B) 0.72 C) 0.2643 D) 0.1 E) 0.0622
25. A fair coin (one for which both the probability of heads and the probability of tails are
0.5) is tossed 100 times. Using binomcdf, the probability that more than 3/5 of the
tosses are heads is about
A) 0.9957 B) 0.33 C) 0.109 D) 0.0176 E) 0.0067
26. A multiple-choice exam has 100 questions, each with five possible answers. If a student
is just guessing at all the answers, the probability that he or she will get 30 or less
correct is about
A) 0.994 B) 0.931 C) 0.875 D) 0.006 E) 0.001
27. A multiple-choice exam has 100 questions, each with five possible answers. If a student
is just guessing at all the answers, the probability that he or she will get at least 30
correct is about
A) 0.994 B) 0.931 C) 0.875 D) 0.0112 E) 0.0001
28. An agricultural researcher plants 30 plots with a new variety of corn. The average yield
for these plots is X = 150 bushels per acre. Assume that the yield per acre for the new
variety of corn follows a normal distribution with unknown mean  and standard
deviation  = 15 bushels. A 95% confidence interval for  is
(A) 150  2 (B) 150  4.5 (C) 150  5.37 (D) 150  7.05
(E) 150  30
29. An agricultural researcher plants 30 plots with a new variety of corn. An 80%
confidence interval for the average yield for these plots is found to be 150  3.51
bushels per acre. Which of the following would produce a confidence interval with a
smaller margin of error than this 80% confidence interval?
A) Choosing a sample with a larger standard deviation
B) Planting 100 plots, rather than 30
C) Choosing a sample with a smaller standard deviation
D) Planting only 5 plots, rather than 30
E) None of the above
30. A private college has a total of 350 students. The Math SAT (SAT-M) score is required
for admission. The mean SAT-M score of all 350 students is 630, and the standard
deviation of SAT-M scores for all 350 students is 40. The formula for a 95% confidence
interval yields the interval 630  5.52. We may conclude that
A) 95% of all student Math SAT scores will be between 624.48 and 635.52.
B) if we repeated this procedure many, many times, only 5% of the 95% confidence
intervals would fail to include the mean SAT-M score of the population of all
students at the college.
C) 95% of the time, the population mean will be between 624.48 and 635.52.
D) the interval is incorrect; it is much too small.
E) none of the above is true.
31. Suppose that the population of the scores of all high school seniors who took the Math
SAT (SAT-M) test this year follows a normal distribution with mean  and standard
deviation  = 100. You read a report that says, “On the basis of a simple random
sample of 100 high school seniors that took the SAT-M test this year, a confidence
interval for  is 512.00  25.76.” The confidence level for this interval is
(A) 90%. (B) 95%. (C) 96%. (D) 99%. (E) none of these.
32. A 95% confidence interval for the mean  of a population is computed from a random
sample and found to be 10  4. We may conclude that
A) there is a 95% probability that  is between 6 and 14.
B) 95% of values sampled are between 6 and 14.
C) if we took many, many additional random samples and from each computed a 95%
confidence interval for , approximately 95% of these intervals would contain .
D) there is a 95% probability that the true mean is 10 and a 95% chance that the true
margin of error is 4.
E) all of the above are true.
33. A 90% confidence interval for the mean  of a population is computed from a random
sample and is found to be 9  4. Which of the following could be the 95% confidence
interval based on the same data?
A) 9  2.
B) 9  3.
C) 9  4.
D) 9  5.
E) Without knowing the sample size, any of the above answers could be the 95%
confidence interval.
Use the following to answer questions 34 through 36:
You measure the heights of a random sample of 400 high school sophomore males in a
Midwestern state. The sample mean is X = 66.2 inches. Suppose that the heights of all high
school sophomore males follow a normal distribution with unknown mean  and standard
deviation  = 4.1 inches.
34. A 90% confidence interval for  (expressed in interval notation) is
A) (58.16, 74.24). B) (59.46, 72.94). C) (65.8, 66.6). D) (65.86, 66.54).
E) (66.18, 66.22).
35. A 99% confidence interval for  (expressed in interval notation) is
A) (58.16, 74.24). B) (66.18, 66.22). C) (65.8, 66.6). D) (64.85, 67.55).
E) (65.67, 66.73).
36. I compute a 99% confidence interval for . Suppose I had measured the heights of a
random sample of 100 sophomore males, rather than 400. Which of the following
statements is true?
A) The margin of error for our 99% confidence interval would increase.
B) The margin of error for our 99% confidence interval would decrease.
C) The margin of error for our 99% confidence interval would stay the same, since the
level of confidence has not changed.
D)  would increase.
E)  would decrease.
37. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures
the motivation, attitude, and study habits of college students. Scores range from 0 to 200
and follow (approximately) a normal distribution with mean 117 and standard deviation
 = 25. You suspect that incoming freshmen have a mean  that is different from 117,
since they are often excited yet anxious about entering college. To test your suspicion,
you test the hypotheses H0:  = 117, Ha:   117. You give the SSHA to an SRS of 50
incoming freshmen and find that their mean score is 118.7. The P-value of your test is
closest to
A) 0.11561. B) 0.23026. C) 0.50521. D) 0.63064. E) 0.81033.
38. The level of calcium in the blood of healthy young adults follows a normal distribution
with mean  = 10 milligrams per deciliter (mg/dl) and standard deviation  = 0.4 mg/dl.
A clinic measures the blood calcium of 100 healthy pregnant young women at their first
visit for prenatal care. The mean of these 100 measurements is X = 9.8 mg/dl. Is this
evidence that the mean calcium level in the population from which these women come
is less than the general population’s mean level of 10 mg/dl? To answer this question,
we test the hypotheses H0:  = 10, Ha:  < 10. The P-value of the test is
A) less than 0.0002. B) 0.0002. C) 0.3085. D) 0.6170. E) greater than 0.99.
39. The nicotine content in milligrams (mg) in cigarettes of a certain brand is normally
distributed with mean  and standard deviation  = 0.1 mg. The brand advertises that
the mean nicotine content of its cigarettes is 1.5 mg, but measurements on a random
sample of 100 cigarettes of this brand gave a mean of X = 1.53 mg. Is this evidence
that the mean nicotine content is actually higher than advertised? To answer this
question, we test the hypotheses H0:  = 1.5, Ha:  > 1.5 at the 5% significance level.
Based on the results, we conclude that
A) H0 should be rejected.
B) H0 should not be rejected.
C) H0 should be accepted.
D) there is a 5% chance that the null hypothesis is true.
E) Ha should be rejected.
40. The time needed for college students to complete a certain paper-and-pencil maze
follows a normal distribution with a mean of 30 seconds and a standard deviation of
3 seconds. You wish to see if the mean completion time  is changed by vigorous
exercise, so you have a group of nine college students exercise vigorously for
30 minutes and then complete the maze. It takes them an average of X = 31.2 seconds
to complete the maze. You use this information to test the hypotheses H0:  = 30, Ha: 
 30 at the 1% significance level. Based on the results, you conclude that
A) H0 should be rejected.
B) H0 should not be rejected.
C) Ha should be accepted.
D) this is a borderline case and no decision should be made.
E) none of these.
41. The time needed for college students to complete a certain paper-and-pencil maze
follows a normal distribution with a mean of 30 seconds and a standard deviation of
3 seconds. You wish to see if the mean time  is changed by vigorous exercise, so you
have a group of 25 college students exercise vigorously for 30 minutes and then
complete the maze. X = 31.2 seconds and you test the hypotheses
H0:  = 30, Ha:   30. You find that the test results are
A) significant at the 5% level.
B) significant at the 1% level.
C) biased.
D) both A) and B) are true.
E) none of the above is true.
42. The water diet requires the dieter to drink two cups of water every half hour from when
he gets up until he goes to bed, but otherwise allows him to eat whatever he likes. 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
Person
1
2
3
4
Weight before the diet
180
125
240
150
Weight after six weeks
170
130
215
152
For the population of all adults, assume that weight loss (in pounds) after six weeks on
the diet (weight before beginning the diet – weight after six weeks on the diet) is
normally distributed with mean . To determine if the diet leads to significant weight
loss, we test the hypotheses H0:  = 0, Ha:  > 0. Based on these data,
A) we would not reject H0 at significance level 0.10.
B) we would reject H0 at significance level 0.10 but not at level 0.05.
C) we would reject H0 at significance level 0.05 but not at level 0.01.
D) we would reject H0 at significance level 0.01 but not at level 0.001.
E) we would reject H0 at significance level 0.001.
43. The weights (in pounds) of three adult males are 160, 215, and 195. The standard error
of the mean of these three weights is
A) 190. B) 27.834. C) 22.73. D) 16.07. E) 13.13.
44. The one sample t statistic from a sample of n = 19 observations for the two-sided test of
H0:  = 6, Ha:   6 has the value t = 1.93. Based on this information, which of the
following would be true?
A) We would reject the null hypothesis at  = 0.10.
B) 0.025 < P-value < 0.05.
C) We would reject the null hypothesis at  = 0.05.
D) Both B) and C) are correct.
E) We would not reject the null hypothesis in a two-sided test, but would reject it in a
one-sided test at  = 0.10.
45. The heights (in inches) of males in the United States are believed to be normally
distributed with mean . The average height of a random sample of 25 American adult
males is found to be X = 69.72 inches, and the standard deviation of the 25 heights is
found to be s = 4.15 inches. The standard error of X is
A) 0.17. B) 0.41. C) 0.69. D) 0.83. E) 2.04.
46. Scores on the Math SAT (SAT-M) are believed to be normally distributed with mean .
The scores of a random sample of three students who recently took the exam are 550,
620, and 480. A 95% confidence interval for  based on these data is
A) 550.00 ± 173.89.
D) 550.00 ± 105.01.
B) 550.00 ± 142.00.
E) 550.00 ± 79.21.
C) 550.00 ± 128.58.
Use the following to answer questions 47 through 50:
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 1 and 2
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.
47. Referring to the information above, assuming two sample t procedures are safe to use, a
99% confidence interval for 1 – 2 (using TI-83/84 two-sample T-Interval) is
A) 4 ± 1.838 yards. B) 4 ± 4.838 yards. C) 4 ± 8.838 yards. D) 4 ± 9.838 yards.
E) 4 ± 13.838 yards.
48. Referring to the information above, suppose the researcher had wished to test the
hypotheses H0: 1 = 2, Ha: 1 > 2. The P-value for the test (using TI-83/84 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.
49. 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.
50. Referring to the information above, if we had used the more accurate TI-83/84
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) 16.559. B) 14.837. C) 12.201. D) 9. E) 8.
Use the following to answer questions 51 through 54:
A random sample of 100 traffic tickets given to motorists in a large city is examined. The tickets
are classified according to the race of the driver. The results are summarized in the following
table.
Number of tickets
White
46
Black
37
Hispanic
11
Other
6
The proportion of this city’s population in each of the racial categories listed above is as follows.
Proportion
White
0.65
Black
0.30
Hispanic
0.03
Other
0.02
We wish to test whether the racial distribution of traffic tickets in the city is the same as the
racial distribution of the city’s population. To do so, we use the χ2 statistic.
51. The component (O – E)2/E of the χ2 statistic corresponding to the category “Hispanic” is
A) 2.67.
B) 5.82.
C) 21.33.
D) 36.51.
E) 4011.36.
52. We compute the value of the χ2 statistic to be 36.52. Assuming that this statistic has
approximately a χ2 distribution, the P-value of our 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.
53. The category that contributes the largest component to the χ2 statistic is
A) White. B) Black. C) Hispanic. D) Other.
E) The answer cannot be determined since this is only a sample.
54. We may assume that the χ2 statistic has an approximate χ2 distribution because of which
of the following?
A) The observed count for each category is greater than 5.
B) The sample size is 100, which is large enough for the approximation to be valid.
C) The expected count for each category is greater than 1.
D) The number of categories is small relative to the number of observations.
E) We may not assume that the χ2 statistic has an approximate χ2 distribution in this
case.
Use the following to answer questions 55 through 59:
You are teaching 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 the students earned the following grades.
Grade
Number
A
56
B
74
C
60
D
9
F
1
You wish to test to see whether the distribution of grades this year was different from the
distribution in the past. To do so, you plan to use the χ2 statistic.
55. The χ2 statistic is approximately
A) 0.2. B) 10.35. C) 33.77.
D) 40.
E) 200.
56. Assuming that the χ2 statistic has approximately a χ2 distribution, how many degrees of
freedom does the distribution have?
A) 200. B) 199. C) 5. D) 4. E) none of these.
57. The grade category that contributes the largest component to the χ2 statistic is
A) A. B) B. C) C. D) D. E) F.
58. The component (O – E)2/E of the χ2 statistic corresponding to a grade of B is about
A) 6.4.
B) 3.27.
C) 33.77.
D) 500.
E) 11,880.30.
59. The P-value of the test is
A) greater than 0.20.
B) between 0.10 and 0.20.
C) between 0.05 and 0.10.
D) between 0.01 and 0.05.
E) less than 0.01.
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