Download Chapter 9: Sampling Distributions

Ch. 9 Review
IB Statistics
*Use normalcdf for all probability problems on this review sheet.
Do NOT use binompdf, binomcdf, geometpdf, or geometcdf.
1. A survey conducted by Raid asked whether the action of a certain type of roach disk would be effective in
killing roaches. Seventy-nine percent of the respondents agreed that the roach disk would be effective.
The number 79% is a
A) statistic. B) variable. C) population. D) parameter. E) biased estimate.
2. A 1993 survey conducted by the local paper in Kansas City, Missouri, one week before election day asked
voters who they would vote for in the City Attorney’s race. Thirty-seven percent said they would for the
Democratic candidate. On election day, 41% actually did vote for the Democratic candidate. The number
41% is a
A) central limit. B) sample. C) population. D) statistic. E) parameter.
3. A phone-in poll conducted by a newspaper reported that 64% of those who called in watched the TV
show South Park on Comedy Central. The number 64% is a(n)
A) unbiased estimate. B) statistic. C) parameter. D) population. E) sample.
4. A phone-in poll conducted by a newspaper reported that 64% of those who called in watched the TV
show South Park on Comedy Central. The unknown true percentage of American citizens who watch
South Park is a
A) statistic. B) variable. C) population. D) parameter. E) sample.
5. The sampling distribution of a statistic is
A) the distribution of values taken by a statistic in all possible samples of the same size from the same
population.
B) the mechanism that determines whether randomization was effective.
C) the probability that we obtain the statistic in repeated random samples.
D) the extent to which the sample results differ systematically from the truth.
E) the distribution of a particular sample of a certain size.
6. The distribution of the values taken on by a statistic in all possible samples from the same population is
called
A) the sample limit theorem.
D) the sampling designation.
B) the sample distribution.
E) the sample designation.
C) the sampling distribution.
7. Brad flips a coin 10 times and records the proportion of heads obtained. He repeats the process of flipping
the coin 10 times and records the proportion of heads obtained many, many times. When done, he makes
a histogram of the results. This histogram represents
A) the bias, if any, that is present.
B) simple random sampling of probabilities.
C) the binomial distribution.
D) the sampling distribution of the proportion of heads in 10 flips of the coin.
E) the true population parameter.
8. A statistic is said to be unbiased if
A) the data was collected from a double-blind study.
B) the median and the mean are equal.
C) the mean of its sampling distribution is equal to the true value of the parameter being estimated.
D) it is only used for honest purposes.
E) the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual
prejudice.
9. If a statistic used to estimate a parameter is such that the mean of its sampling distribution is equal to the
true value of the parameter being estimated, the statistic is said to be
A) random. B) unbiased. C) biased. D) normal. E) a proportion.
10. The variability of a statistic is described by
A) the spread of its sampling distribution.
B) the amount of bias present.
C) the different values the statistic will take from each of the samples.
D) the stability of the population it describes.
E) the vagueness in the wording of the question used to collect the sample data.
11. A simple random sample of 50 undergraduates at Auburn University found that 60% of those sampled felt
that drinking was a problem among college students. A simple random sample of 50 undergraduates at
The University of Southern California found that 60% felt that drinking was a problem among college
students. The number of undergraduates at Auburn is approximately 5,000 while the number at USC is
approximately 40,000. We conclude that
A) the sample from USC has about 12.5% more variability than that from Auburn because the
population size was much larger.
B) the sample from Auburn has much more variability than that from USC.
C) the sample from Auburn has much less variability than that from USC.
D) the sample from Auburn has the same variability as that from USC because the sample sizes were
the same.
E) it is impossible to make any statements about the variability of the two samples since the students
surveyed were different.
12. The number of undergraduates at Auburn University is approximately 5,000 while the number at USC is
approximately 40,000. A simple random sample of about 3% of the undergraduates is taken from both
schools. We conclude that
A) the sample from Auburn has less variability than that from USC.
B) the sample from Auburn has more variability than that from USC.
C) the sample from Auburn has about the same variability as that from USC.
D) it is impossible to make any statements about the variability of the two samples since the students
surveyed were different.
E) the sample from Auburn has 0.03 times more variability than that from USC.
13. Suppose you are going to roll a die 60 times and record p̂ , the proportion of times that an even number
(2, 4, or 6) is showing. The sampling distribution of p̂ should be centered at about
A) 1/2. B) 1/3. C) 1/6. D) 5/6. E) 1/36.
Use the following to answer questions 14-16:
A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to
6%, with the additional revenue going to education. Let p̂ denote the proportion in the sample that say they support
the increase. Suppose that 30% of all adults in Ohio support the increase.
14. The mean
A) 0.30.
 p̂
of p̂ is
B) 30% ± 5%.
15. The standard deviation
A) 0.05
B) 0.0118.
 p̂
C) 30% ± 2.5%.
D) 450.
E) 5%
of p̂ is
C) 0.0126.
D) 0.40.
E) 0.00014
16. The probability that p̂ is more than 0.35 is
A) less than 0.001. B) about 1. C) .9441.
D) 0.40.
E) 0.0559.
17. A fair coin (one for which both the probability of heads and the probability of tails are 0.5) is tossed 60
times. The probability that 1/3 or less of the tosses are heads is about
A) 0.9957. B) 0.33. C) 0.109. D) 0.09. E) 0.0049.
18. 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) 0.01. B) 0.099. C) 1/9. D) 0.05. E) 0.0009.
19. 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 30 correct if the subject does not have ESP
and is just guessing? *Hint: Use the percentages, and do normalcdf(0.30, 1, 0.25, 0.0433).
A) 0.310. B) 0.250. C) 0.1241. D) 0.043. E) less than 0.001.
20. 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 less than 30 correct is
A) 0.2500. B) 0.1230. C) 0.1056. D) 0.0062. E) 0.9938.
21. A college basketball player makes 75% of his free throws. Over the course of the season he will attempt
1000 free throws. Assuming free throw attempts are independent, the probability that the number of free
throws he makes exceeds 750 is approximately
A) 0.2. B) 0.2266. C) 0.6454. D) 0.7734. E) 0.5.
22. A college basketball player makes 80% of his free throws. Over the course of the season he will attempt
100 free throws. Assuming free throw attempts are independent, what is the approximate probability that
he makes at least 90 of these attempts?
A) 0.0062. B) 0.72. C) 0.2643. D) 0.10. E) 0.90.
23. A random sample of size 100 is to be taken from a population that is normally distributed with mean 2500
and standard deviation 10. The average
X of the observations in our sample is to be computed. The
sampling distribution of X is
A) normal with mean 2500 and standard deviation 1.
B) normal with mean 2500 and standard deviation 10.
C) normal with mean 2500 and standard deviation 0.4.
D) normal with mean 2500 and standard deviation 3.14.
E) normal with mean 25 and standard deviation 1.
24. A random sample of size 25 is to be taken from a population that is normally distributed with mean 60
and standard deviation 10. The number X of the observations in our sample that are larger than 60 is to be
computed. The sampling distribution of X is
A) normal with mean 60 and standard deviation 10.
B) normal with mean 60 and standard deviation 25.
C) uniform.
D) skewed to the left.
E) none of the above.
25. An automobile insurer has found that repair claims have a mean of $920 and a standard deviation of $870.
Suppose that the next 100 claims can be regarded as a random sample from the long-run claims process.
The mean and standard deviation of the average X of these 100 claims are
A) mean = $920 and standard deviation = $870.
B) mean = $920 and standard deviation = $2.95.
C) mean = $920 and standard deviation = $87.
D) mean = $92 and standard deviation = $87.
E) mean = $92 and standard deviation = $870.
26. In a large population of adults, the mean IQ is 112 with a standard deviation of 20. Suppose 200 adults are
randomly selected for a market-research campaign. The distribution of the sample mean IQ is
A) impossible to compute
B) exactly normal with mean 112 and standard deviation 1.414.
C) exactly normal with mean 112 and standard deviation 20.
D) approximately normal with mean 112 and standard deviation 1.414.
E) approximately normal with mean 112 and standard deviation 20.
27. A random variable X has mean  X and standard deviation X. Suppose n independent
observations
of X are taken and the average x of these n observations is computed. We can assert that if n is very
large, the sampling distribution of x is approximately normal. This assertion follows from:
A)
the law of large numbers.
B)
the bell curve.
C)
the definition of a sampling distribution.
D)
the central limit theorem.
E)
the standard deviation of the sampling distribution.
Use the following to answer questions 28 through 30:
The scores of individual students on the American College Testing (ACT) Program composite college entrance
examination have a normal distribution with mean 18.6 and standard deviation 6.0. At Northside High, 36 seniors
take the test.
28. If the scores at this school have the same distribution as national scores, what is the mean of the sampling
distribution of the average (sample mean) score for the 36 students?
A) 18.6. B) 3.1. C) 6.0. D) 1.0.
E) This value cannot be determined without the actual data.
29. If the scores at this school have the same distribution as national scores, what is the standard deviation of
the sampling distribution?
A) 0.41. B) 3.1. C) 1.0. D) 6.0. E) none of these.
30. If the scores at this school have the same distribution as national scores, the shape of the sampling
distribution is
A) approximately normal, but the approximation is poor.
B) approximately normal, and the approximation is good.
C) skewed right.
D) neither normal nor non-normal. It depends on the particular 36 students selected.
E) exactly normal.
31. I select a simple random sample of 5000 batteries produced in a manufacturing plant. I test each battery
X of all 5000 failure
X might be modeled as having approximately a
and record how long it takes for each battery to fail. I then compute the average
times. The sampling distribution of
A) geometric distribution.
B) normal distribution.
C) uniform distribution.
D)
E)
binomial distribution.
none of the above.
32. Suppose that you are a student worker in the statistics department, and they agree to pay you using the
Random Pay system. Each week the chair of the department flips a coin. If it comes up heads, your pay
for the week is $80; if it comes up tails, your pay for the week is $40. Your friend is working for the
engineering department and makes $65 per week. The probability that your total earnings in 100 weeks
are more than hers is approximately
A) 0. B) 0.1586. C) 0.4013. D) 0.5000. E) 0.5987
33. A researcher initially plans to take a SRS of size n from a population that has mean 80 and standard
deviation 20. If he were to double his sample size (to 2n), the standard deviation of the sampling
distribution of
A) 1/
2.
X would change by a factor of
B) 2. C) 1/2. D) 1/ 2n .
E) 1.
34. The weights of extra-large eggs have a normal distribution with a mean of 1 ounce and a standard
deviation of 0.1 ounces. The probability that a dozen extra-large eggs has a total weight of more than 13
ounces is closest to
A) 0. B) 0.1028. C) 0.1814. D) 0.2033. E) 0.9982.
35. The incomes in a certain large population of college teachers have a normal distribution with mean
$35,000 and standard deviation $5000. Four teachers are selected at random from this population to serve
on a salary review committee. What is the probability that their average salary exceeds $40,000?
A) Essentially 0. B) 0.0228. C) 0.1587. D) 0.3413. E) 0.9772.
Use the following to answer questions 36 and 37:
The distribution of actual weights of 8-ounce chocolate bars produced by a certain machine is normal with mean 8.1
ounces and standard deviation 0.1 ounces.
36. If a sample of five of these chocolate bars is selected, the probability that their average weight is less than
8 ounces is about
A) 0.0127. B) 0.1853. C) 0.2389. D) 0.4871. E) 0.9873.
37. If a sample of five of these chocolate bars is selected, there is only a 5% chance that their average weight
will be below
A) 7.94 ounces. B) 8.03 ounces. C) 8.08 ounces. D) 8.17 ounces.
E) 8.20 ounces.
Use the following to answer questions 38 and 39:
The SAT scores of entering freshmen at University X have a N(1200, 90) distribution, and the SAT scores of
entering freshmen at University Y have a N(1215, 110) distribution. A random sample of 100 freshmen is sampled
from each university. Let X = the sample mean of the 100 scores from University X and
the 100 scores from University Y.
38. The probability that X is less than 1190 is about
A) 0.0116. B) 0.1335. C) 0.4090. D) 0.4562.
E) 0.8665.
Y = the sample mean of
39. The probability that Y is less than 1190 is about
A) 0.0116. B) 0.1335. C) 0.4090. D) 0.4562.
E) 0.9884.
Use the following to answer questions 40 and 41:
A factory produces plate-glass sheets with a mean thickness of 4 millimeters and a standard deviation of thickness of
1.1 millimeters. A simple random sample of 100 sheets of glass is to be selected and measured, and the sample mean
thickness
X of the 100 sheets is to be computed.
40. We know the random variable X has approximately a normal distribution because of
A) the law of large numbers.
B) the fact that the standard deviation is less than 50% of the mean.
C) the law of proportions.
D) the fact that probability is the long run proportion of times an event occurs.
E) the central limit theorem.
41. The probability that the average thickness X of the 100 sheets of glass is less than 4.1 millimeters is
approximately
A) 0.8183. B) 0.6817. C) 0.3268. D) 0.3183.
E) 0.1817.
42. In a large population of adults, the mean IQ is 112 with a standard deviation of 20. Suppose 200 adults are
randomly selected for a market-research campaign. The probability that the sample mean IQ of these
adults is greater than 110 is approximately
A) 0.5398. B) 0.799. C) 0.021. D) 0.418. E) 0.9214.
43. An automobile insurer has found that repair claims have a mean of $920 and a standard deviation of $870.
Suppose that the next 100 claims can be regarded as a random sample from the long-run claims process.
The probability that the average X of these 100 claims is larger than $1000 is approximately
A) 0.9200. B) 0.8212. C) 0.3600. D) 0.1789. E) 0.
44. Grace forgot to study for her biology final exam and has no idea what will be on it. The final exam is
multiple-choice and has 100 questions, each with five possible answers. If Grace just guesses on every
question, the probability that she will get 25 or more correct is about
A) 0.2500. B) 0.1230. C) 0.1056. D) 0.0062. E) 0.
45. Chase neglected to study for an IB statistics test and plans to put “C” for every multiple choice question.
The multiple-choice test has 45 questions, each with five possible answers. Chase’s expected value for
number of correct answers is
A) 1. B) 5. C) 9. D) 15. E) 45.
46. Refer to #45. The probability that Chase will pass the test (using this strategy) is close to
A) 0. B) 0.1030. C) 0.1553. D) 0.6200. E) 0.9938.
47. Loren plans to put “A” for every multiple choice question on the next IB biology test. She knows that the
multiple-choice test will have 75 questions, each with four possible answers. Loren’s expected value for
number of correct answers is closest to
A) 1. B) 5. C) 9. D) 19. E) 40.
48. Refer to #47. The probability that Loren will not pass the test (using this strategy) is close to
A) 1. B) 0.5000. C) 0.3918. D) 0.0231. E) 0.0001.
49. Shawn plans to guess on every multiple choice question on the ACT. This year’s ACT is a multiplechoice test that has 340 questions, each with four possible answers. Shawn’s expected value for number of
correct answers is
A) 11. B) 15. C) 25. D) 55. E) 85.
50. Refer to #49. The probability that Shawn will get at most 95 questions correct (using this strategy) is
about
A) 0.1052. B) 0.4521. C) 0.3762. D) 0.8948. E) 0.9999.
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