Chap 6, 7, 8
39. A sample is obtained from a population with u = 100 and o=20. Which of the following samples would produce the most extreme z-score? a. A sample of n=25 scores with M = 102 b. A sample of n= 100 scores with M= 102 c. A sample of n=25 scores with M = 104 d. A sample of n= 100 scores with M=104 ANSWER: d
D
53. The mean for a sample of n=9 scores has a standard error of 2 points. This sample was selected from a population with a standard deviation of s = 18. a. True b. False ANSWER: False
F
53. You can reduce the risk of a Type I error by using a larger sample. a. True b. False
F
54. A vertical line is drawn through a normal distribution dividing the distribution into two parts. If the smaller part equals 0.1915 of the distribution, then the line was drawn at z=0.50 or at z=-0.50. a. True b. False
F
56. The smallest possible standard error is obtained when a small sample is taken from a population with a small standard deviation. a. True b. False ANSWER: False
F
57. On average, a sample of n= 16 scores from a population with s = 10 will provide a better estimate of the population mean than you would get with a sample of n= 16 scores from a population with s = 5. a. True b. False ANSWER: False
F
58. A sample is obtained from a population with s = 20. If the sample mean has a standard error of 5 points, then the sample size is n=4. a. True b. False ANSWER: False
F
59. A researcher administers a treatment to a sample from a population with a mean of m= 60. If the treatment is expected to increase scores and a one-tailed test is used to evaluate the treatment effect, then the null hypothesis states that m 60. a. True b. False
F
59. If the sample size is equal to the population standard deviation (n=s), then the standard error for the sample mean is equal to 1.00. a. True b. False ANSWER: False
F
61. If one score is randomly selected from a normal distribution with u = 100 and o=20, the probability of obtaining a score less than X = 70 is p=0.0013. a. True b. False
F
62. A researcher administers a treatment to a sample of n= 16 selected from a population with m= 40 and s = 8. If the sample mean after treatment is M = 42, then Cohen's d= 1.00. a. True b. False
F
62. A sample of n=9 scores is randomly selected from a population with u = 80 and s=9. If the sample mean is M=83, then the corresponding z-score is z= +3.00. a. True b. False ANSWER: False
F
63. A sample of n= 4 scores is selected from a population with u = 30 and s = 8. The probability of obtaining a sample mean greater than 34 is equal to the probability of obtaining a z-score greater than z=2.00 from a normal distribution. a. True b. False ANSWER: False
F
63. If one score is randomly selected from a normal distribution with u = 100 and 6 = 20, the probability of obtaining a score between X = 80 and X = 120 is p=0.3413. a. True b. False
F
66. A sample of n=25 scores is selected from a population with u = 50 and o= 10. The probability of obtaining a sample mean greater than 55 is p=0.3085.
F
68. A sample of n=25 scores is selected from a population with u = 70 and o= 20. It is very unlikely that the sample mean will be smaller than 72. a. True b. False
F
69. A population has u = 60 and 6 = 30. For a sample of n = 25 scores from this population, a sample mean of M=55 would be considered an extreme value. a. True b. False
F
70. For the normal approximation to the binomial distribution with n=100 and p=1/5, the probability of selecting a score greater than or equal to 25 is p=0.1056. a. True b. False
F
1. What term is used to identify the mean of the distribution of sample means? a. the expected value of M b. the standard error of M c. the sample mean d. the central limit mean
A
12. What is the relationship between the alpha level, the size of the critical region, and the risk of a Type I error? a. As the alpha level increases, the size of the critical region increases and the risk of a Type I error increases. b. As the alpha level increases, the size of the critical region increases and the risk of a Type I error decreases. c. As the alpha level increases, the size of the critical region decreases and the risk of a Type I error increases. d. As the alpha level increases, the size of the critical region decreases and the risk of a Type I error decreases.
A
13. Even if a treatment has no effect, it is still possible to obtain an extreme sample mean that is very different from the population mean. What outcome is likely if this happens? a. Reject Hy and make a Type I error. b. Correctly reject Ho. c. Fail to reject H, and make a Type II error. d. Correctly fail to reject Ho.
A
42. In general, the null hypothesis states that the treatment has no effect on the population parameter being studied. a. True b. False
T
42. The sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. a. True b. False ANSWER: True
T
43. The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. a. True b. False ANSWER: True
T
44. A sample of n = 25 scores is selected from a population with a mean of u = 80 and a standard deviation of o= 20. The expected value for the sample mean is 80. a. True b. False ANSWER: True
T
44. A vertical line drawn through a normal distribution at z= 1.25 will separate the distribution into two sections. The proportion in the smaller section is 0.1056. a. True b. False
T
44. Most researchers would like the hypothesis test to reject the null hypothesis. a. True b. False
T
45. The critical region for a hypothesis test consists of sample outcomes that are very unlikely to occur if the null hypothesis is true. a. True b. False
T
46. A vertical line drawn through a normal distribution at z=-0.80 will separate the distribution into two sections. The proportion in the larger section is 7881. a. True b. False
T
47. When the Z-score value in a normal distribution is negative, the majority of the area is on the right-hand side of the distribution a. True b. False
T
48. For any normal distribution, the proportion in the tail above z=2.00 is p=0.0228. a. True b. False
T
48. If samples are selected from a normal population, the distribution of sample means will also be normal. a. True b. False ANSWER: True
T
48. If the sample data are in the critical region with a =.01, then the same sample data would still be in the critical region if a were changed to .05. a. True b. False
T
49. According to the central limit theorem, the standard error for a sample mean becomes smaller as the sample size increases. a. True b. False ANSWER: True
T
49. For a normal distribution, the proportion in the tail below z=-2.00 is equal to 0.0228. a. True b. False
T
50. A Type I error occurs when a treatment has no effect but the decision is to reject the null hypothesis. a. True b. False
T
50. For any normal distribution, exactly 97.50% of the z-score values are less than z= 1.96. a. True b. False
T
50. If samples of size n= 16 are selected from a population with u = 40 and o=8, the distribution of sample means will have an expected value of 40. a. True b. False ANSWER: True
T
51. A Type I error occurs when a researcher concludes that a treatment has an effect but, in fact, the treatment has no effect. a. True b. False
T
51. For a normal distribution, the proportion in the tail beyond z=1.50 is p=0.0668. a. True b. False
T
51. If samples of size n= 16 are selected from a population with u = 40 and o=8, the distribution of sample means will have a standard error of 2 points. a. True b. False ANSWER: True
T
52. A mathematical proposition known as the central limit theorem provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean. a. True b. False ANSWER: True
T
52. The alpha level determines the risk of a Type I error. a. True b. False
T
54. The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.. a. True b. False ANSWER: True
T
54. There is always a possibility that the decision reached in a hypothesis test is incorrect. a. True b. False
T
55. A research report includes the statement, z=2.13, p<.05. For this hypothesis test, the used null hypothesis is rejected using an alpha level of a =.05. a. True b. False
T
55. If the standard deviation for a population increases, the standard error for sample means from the population will also increase. a. True b. False ANSWER: True
T
56. In a research report, the notation p<.05 indicates that the probability of a Type I error is less than .05. a. True b. False
T
57. For a population with a mean of u = 80 and 6 = 10, only 2.28% of the scores are greater than X = 100. a. True b. False
T
57. If a research report includes the term significant result, it means that the null hypothesis was rejected. a. True b. False
T
58. For a normal distribution with u = 100 and o=20, the score that separates the top 60% of the distribution from the bottom 40% is X=95. a. True b. False
T
58. If the research prediction is that the treatment will decrease scores, then the critical region for a directional test will be in the left-hand tail. a. True b. False
T
60. A researcher is evaluating a treatment that is expected to increase scores. If a one-tailed test with a = .05 is used, then the critical region consists of z-scores greater than 1.65. a. True b. False
T
60. A researcher obtained M= 27 for a sample of n= 36 scores selected from a population with u = 30 and s = 18. This sample mean corresponds to a z-score of z=-1.00. a. True b. False ANSWER: True
T
61. A directional hypothesis test is also referred to as a one-tailed test. a. True b. False
T
61. A sample of n=9 scores is selected from a normal population with a mean of u = 80 and a standard deviation of s = 12. The probability that the sample mean will be greater than M=86 is equal to the probability of obtaining a z-score greater than z= 1.50. a. True b. False ANSWER: True
T
63. A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect. a. True b. False
T
64. A sample of n=4 scores is selected from a population with u = 70 and s = 10. The probability of obtaining a sample mean greater than 65 is p = 0.8413. a. True b. False ANSWER: True
T
64. The value obtained for Cohen's d is independent of the sample size. a. True b. False
T
65. A sample of n=9 scores is selected from a population with u = 50 and s = 12. The probability of obtaining a sample mean greater than 46 is p = 0.8413.Z
T
65. A significant treatment effect does not necessarily indicate a large treatment effect. a. True b. False
T
65. The binomial distribution for p= 1/4 and n=96 has a mean of u = 24. a. True b. False
T
66. One of the simplest and most direct methods for measuring effect size is Cohen's d. a. True b. False
T
67. A sample of n= 16 scores is selected from a population with u = 70 and o= 8. It is very unlikely that the sample mean will be greater than 78.
T
67. The power of a hypothesis test is the probability that the sample mean will be in the critical region if the treatment has an effect. a. True b. False
T
68. If the power for a hypothesis test is calculated to be 0.80, then for same test, the probability of a Type II error is 0.20. a. True b. False
T
69. If all other factors are held constant, increasing the sample size from n= 25 to n= 100 will increase the power of a statistical test. a. True b. False
T
70. A population has u = 60 and o= 10. For a sample of n=25 scores from this population, a sample mean of M= 55 would be considered an extreme value. a. True b. False
T
27. For a normal population with a mean of u = 80 and a standard deviation of o= 10, what is the probability of obtaining a sample mean greater than M = 75 for a sample of n = 25 scores? a. p=0.0062 b. p=0.9938 c. p=0.3085 d. p=0.6915 ANSWER: b
B
27. Which of the following accurately describes the effect of increasing the sample size? a. Increases the standard error and has no effect on the risk of a Type I error b. Decreases the standard error and has no effect on the risk of a Type I error c. Increases the risk of a Type I error and has no effect on the standard error d. Decreases the risk of a Type I error and has no effect on the standard error
B
29. A researcher administers a treatment to a sample of participants selected from a population with u = 80. If the researcher obtains a sample mean of M = 88, given the same sample size, which combination of factors is most likely to result in rejecting the null hypothesis? a. o=5 and a = .01 b. o = 5 and a = .05 c. o= 10 and a=.01 d. o= 10 and a=.05
B
30. A researcher administers a treatment to a sample of participants selected from a population with u = 80. If the researcher obtains a sample mean of M = 88, given the same alpha level, which combination of factors is most likely to result in rejecting the null hypothesis? a. o = 5 and n=25 b. o = 5 and n=50 c. 0= 10 and n=25 d. o= 10 and n= 50
B
31. A sample of n= 4 scores is obtained from a population with u = 70 and o=8. If the sample mean corresponds to a z score of 2.00, then what is the value of the sample mean? a. M= 86 b. M= 78 c. M= 74 d. M= 72 ANSWER: b
B
32. A sample from a population with u = 40 and o= 10 has a mean of M= 44. If the sample mean corresponds to a z= 2.00, then how many scores are in the sample? a. n= 100 b. n=25 c. n=5 d. n=4 ANSWER: b
B
33. A random sample of n= 16 scores is obtained from a population with o= 12. If the sample mean is 6 points greater than the population mean, what is the z-score for the sample mean? a. +6.00 b. +2.00 c. +1.00 d. It cannot be determined without knowing the population mean. ANSWER: b
B
36. A sample of n= 16 individuals is selected from a population with u = 60 and o=6 and a treatment is administered to the sample. After treatment, the sample mean is M=63. What is the value of Cohen's d for this sample? a. 0.33 b. 0.50 c. 2.00 d. 3.00
B
38. A multiple-choice test with 48 questions has four choices for each question. What is the probability of getting more than 12 questions correct by just guessing? a. 0.5000 b. 0.4325 c. 0.5675 d. 0.0675
B
38. A random sample of n= 16 scores is obtained from a normal population with u = 40 and o= 8. What is the probability that the sample mean will be within 2 points of the population mean? a. 0.3830 b. 0.6826 c. 0.8664 d. 0.9544 ANSWER: b
B
4. What proportion of a normal distribution is located in the tail below z=2.00? a. 0.9772 b. 0.0228 c. 0.4772 d. 0.0456
B
6. A two-tailed hypothesis test is being used to evaluate a treatment effect with a = .05. If the sample data produce a z score of z = 2.24, then what is the correct decision? a. Reject the null hypothesis and conclude that the treatment has no effect. b. Reject the null hypothesis and conclude that the treatment has an effect. c. Fail to reject the null hypothesis and conclude that the treatment has no effect. d. Fail to reject the null hypothesis and conclude that the treatment has an effect.
B
7. The critical boundaries for a hypothesis test are z=+1.96 and -1.96. If the Z-score for the sample data is z=1.90, then what is the correct statistical decision? a. Fail to reject Hi b. Fail to reject H. c. Reject H d. Reject H
B
21. For a particular population a sample of n= 4 scores has an expected mean value of 10. For the same population, a sample of n=25 scores would have an expected mean value of a. 4 b. 8 c. 10 d. 20
C
22. What is the consequence of a Type II error? a. Concluding that a treatment has an effect when it really does b. Concluding that a treatment has no effect when it really has no effect c. Concluding that a treatment has no effect when it really does d. Concluding that a treatment has an effect when it really has no effect
C
23. A normal distribution has a mean of u = 70 with o= 12. If one score is randomly selected from this distribution, what is the probability that the score will be less than X= 76? a. 0.1915 b. 0.3085 c. 0.6915 d. 0.3830
C
24. A sample of n= 4 scores has a standard error of 10 points. For the same population, what is the standard error for a sample of n= 16 scores? a. 1 b. 2.5 c. 5 d. 10 ANSWER: C
C
24. When is there a risk of a Type II error? a. Whenever Ho is rejected b. Whenever H is rejected c. Whenever the decision is "fail to reject H." d. The risk of a Type II error is independent of the decision from a hypothesis test.
C
25. A random sample of n=4 scores is obtained from a population with a mean of u = 80 and a standard deviation of o= 10. If the sample mean is M = 90, what is the z-score for the sample mean? a. z= 20.00 b. z=5.00 c. z=2.00 d. z= 1.00 ANSWER: c.
C
15. What z-score value separates the top 10% of a normal distribution from the bottom 90%? a. z= 1.28 b. z=0.25 c. Z=-1.28 d. z=-0.25
A
18. Which of the following represents the probability of a Type II error? b. o c. B d. a ANSWER: C
C
37. A treatment is administered to a sample of n=9 individuals selected from a population with a mean of u = 80 and a standard deviation of o= 12. After treatment, the effect size is measured by computing Cohen's d, and a value of d= 0.50 is obtained. Based on this information, what is the mean for the treated sample? a. M = 6 b. M=82 c. M=86 d. This cannot be answered without knowing the sample size.
C
70. If other factors such as sample size, alpha level, and population standard deviation are held constant, the larger the size of the treatment effect, the greater the power of the hypothesis test. a. True b. False
T
24. A normal distribution has a mean of u = 70 with o = 12. If one score is randomly selected from this distribution, what is the probability that the score will be less than X= 55? a. 0.3944 b. 0.1056 c. 0.8944 d. 0.7888
B
28. A normal distribution has a mean of u = 80 with o = 20. What score separates the highest 15% of the distribution from the rest of the scores? a. X= 59.2 b. X= 100.8 c. X=95 d. X = 65
B
29. A normal distribution has a mean of u = 80 with o=20. What score separates the highest 40% of the distribution from the rest of the scores? a. X= 75 b. X=85 c. X= 54.4 d. X= 105.6
B
3. A jar contains 10 red marbles and 30 blue marbles. A random sample without replacement of n=3 marbles is selected from the jar. If the first two marbles are both blue, what is the probability that the third marble will be red? a. 10/37 b. 10/38 c. 10/40 d. 8/38
B
3. For a population with u = 80 and o=20, the distribution of sample means based on n= 16 will have an expected value of_ and a standard error of a. 5; 80 b. 80; 5 c. 20; 20 d. 80; 1.25
B
30. A normal distribution has a mean of u = 80 with o=20. What score separates the lowest 30% of the distribution from the rest of the scores? a. X=90.4 b. X= 69.6 c. X=110 d. X = 50
B
31. A normal distribution has a mean of u = 24 with o=3. What is the minimum score needed to be in the top 14% of the distribution? a. X= 20.76 b. X = 27.24 c. X=25.08 d. X= 24.42
B
40. A true/false test has 100 questions. Using the normal approximation to the binomial distribution, what is the probability of getting more than 55 correct by just guessing? a. p(X> 55) b. p(X> 55.5) c. p(X> 54.5) d. p(X> 45)
B
5. What proportion of a normal distribution is located in the tail beyond z=-1.00? a. 0.8413 b. 0.1587 c. 0.3413 d. 0.1587
B
6. A sample of n= 16 scores is selected from a population with u = 80 with o = 20. On average, how much error would be expected between the sample mean and the population mean? a. 20 points b. 5 points c. 4 points d. 1.25 points
B
9. A vertical line is drawn through a normal distribution at z=0.80. What proportion of the distribution is on the right hand side of the line? a. 0.7881 b. 0.2119 c. 0.2881 d. 0.5762
B
52. For a normal distribution, the proportion located between z=-1.00 and z= +1.00 is p= 34.13%. a. True b. False
F
53. For any normal distribution, the proportion located between the mean and z=1.40 is 0.9192. a. True b. False
F
15. Which of the following correctly describes the effect of increasing the alpha level (for example, from .01 to .05)? a. Increase the likelihood of rejecting Ho and increase the risk of a Type I error. b. Decrease the likelihood of rejecting Ho and increase the risk of a Type I error. c. Increase the likelihood of rejecting H, and decrease the risk of a Type I error. d. Decrease the likelihood of rejecting Ho and decrease the risk of a Type I error.
A
23. When is there a risk of a Type I error? a. Whenever H, is rejected b. Whenever H, is rejected c. Whenever the decision is "fail to reject H." d. The risk of a Type I error is independent of the decision from a hypothesis test.
A
3. What is measured by the denominator of the z-score test statistic? a. The average distance between M and u that would be expected if Hy was true b. The actual distance between M and u c. The position of the sample mean relative to the critical region d. Whether or not there is a significant difference between M and u
A
33. A researcher expects a treatment to produce an increase in the population mean. The treatment is evaluated using a one tailed hypothesis test, and the test produces z= +1.85. Based on this result, what is the correct statistical decision? a. The researcher should reject the null hypothesis with a = .05 but not with a = .01. b. The researcher should reject the null hypothesis with either a =.05 or a=.01. c. The researcher should fail to reject Hy with either a=.05 or a=.01. d. The researcher should change a.
A
34. A researcher is conducting an experiment to evaluate a treatment that is expected to increase the scores for individuals in a population. If the researcher uses a one-tailed test with a = .01, then which of the following correctly identifies the critical region? a. z>2.33 b. z>2.58 c. z<2.33 d. z<2.58
A
34. For a normal population with u = 40 and o= 10 which of the following samples is least likely to be obtained? a. M=42 for a sample of n=4 b. M = 44 for a sample of n=4 c. M=42 for a sample of n= 100 d. M=44 for a sample of n= 100 ANSWER: a
A
35. Under what circumstances can a very small treatment effect still be significant? a. If the sample size (n) is very large b. If the sample standard deviation (6) is very large c. If the standard error of M(O m) is very large d. All of the other three factors will result in a significant treatment effect.
A
38. If a hypothesis test is found to have power=0.70, then what is the probability that the test will result in a Type II error? a. 0.30 b. 0.70 c. p>0.70 d. 0.40
A
4. Which of the following accurately describes the critical region? a. Outcomes with a very low probability if the null hypothesis is true b. Outcomes with a high probability if the null hypothesis is true c. Outcomes with a very low probability whether or not the null hypothesis is true d. Outcomes with a high probability whether or not the null hypothesis is true
A
8. A researcher conducts a hypothesis test to evaluate the effect of a treatment. The hypothesis test produces a z-score of z=2.37. Assuming that the researcher is using a two-tailed test, what decision should be made? a. The researcher should reject the null hypothesis with a = .05 but not with a=.01. b. The researcher should reject the null hypothesis with either a = .05 or a=.01. c. The researcher should fail to reject Hy with either a=.05 or a=.01. d. The researcher should ignore the results.
A
74. A researcher is testing the effectiveness of a new herbal supplement that claims to improve physical fitness. A sample of n= 16 college students is obtained and each student takes the supplement daily for six weeks. At the end of the 6 week period, each student is given a standardized fitness test and the average score for the sample is M = 39. For the general population of college students, the distribution of test scores is normal with a mean of u = 35 and a standard deviation of o = 12. Do students taking the supplement have significantly better fitness scores? Use a one-tailed test with a=.05.
ANSWER: Ho: u <35 (no increase). H :u> 35. The critical region consists of Z-scores in the right-hand tail beyond z=+1.65. The standard error is 3.00 and z= 1.33. Fail to reject Ho. The sample data is not sufficient to conclude that the supplement has a significant effect on improving physical fitness.
76. A researcher selects a sample of n= 25 from a normal population with u = 40 and o = 10. If the treatment is expected to increase scores by 3 points, what is the power of a two-tailed hypothesis test using a = .05?
ANSWER: The critical region boundary of z=+1.96 corresponds to a sample mean of M = 43.92. If the treatment has a 3-point effect, then the distribution of sample means will be centered at m= 43 and the boundary of M= 43.92 will have a z-score of z=0.46. Power = p<z> 0.46) = 0.3228.
71. Define the critical region for a hypothesis test, and explain how the critical region is related to the alpha level.
ANSWER: The critical region consists of sample outcomes that are very unlikely to occur if the null hypothesis is true. The value of alpha is used to define a precise probability for the term very unlikely.
71. Define the distribution of sample means.
ANSWER: The distribution of sample means is the set of sample means obtained from all the possible random samples of a specified size (n) taken from a particular population.
72. Describe what is measured by the standard error of M.
ANSWER: The standard error of M is the standard deviation of the distribution of sample means, and provides a measure the expected distance (or deviation) on average between a sample mean M and the population mean u.
75. Some researchers claim that herbal supplements such as ginseng or ginkgo biloba enhance human memory. To test this claim, a researcher selects a sample of n=25 college students. Each student is given a ginkgo biloba supplement daily for six weeks and then all the participants are given a standardized memory test. For the population, scores on the test are normally distributed with u = 70 and o= 15. The sample of n=25 students had a mean score of M= 75. a. Are the data sufficient to that the herb has a significant effect on memory? Use a two-tailed test with a =.05. b. Compute Cohen's d for this study.
ANSWER: a. HO: u <= 70 (students who take the herb are no different from the population). With a= .05, the critical region consists of z-scores beyond +1.96. The standard error is oM = 3 and z= 1.67. Fail to reject H0 and conclude that the herb has no significant effect in enhancing memory. b. Cohen's d=5/15 = 0.33.
72. The term error is used in two different ways in hypothesis testing: a Type I error (or Type II) and the standard error. a. What can a researcher do to influence the size of the standard error? Does this action have any effect on the probability of a Type I error? b. What can a researcher do to influence the probability of a Type I error? Does this action have any effect on the size of the standard error?
ANSWER: a. Increasing the sample size decreases the standard error. Changes in sample size have no effect on the probability of a Type I error. b. The probability of a Type I error is under the direct control of the researcher who selects the alpha level for the test. The choice of an alpha level does not affect the standard error.
76. A sample of n = 16 scores is selected from a normal population with u = 40 and o= 12. a. Describe the distribution of sample means that contains the mean for the sample. b. What is the probability that the sample mean will be greater than 43? c. What is the probability that the sample mean will be less than 34?
ANSWER: a. The distribution will be normal (because the underlying population has a normal distribution), has an expected mean value of u = 40 and a standard error of 12/4 = 3 points. b. z=1.00, p=0.1587 c. Z=-2.00, p=0.0228
73. Describe the shape, the mean, and the standard deviation for each of the following two distributions. a. A population of scores with u = 50 and o=6. b. The distribution of sample means based on samples of n= 36 selected from a population with u = 50 and o=6.
ANSWER: a. The population is a distribution of scores with an unknown shape, a mean of u = 50 and a standard deviation of o=6. b. The distribution of sample means is approximately normal (because n > 30), with a mean (expected value) of u = 50, and a standard deviation (standard error) of OM=1.
73. A researcher would like to determine whether a new tax on cigarettes has had any effect on people's behavior. During the year before the tax was imposed, stores located in rest areas on the state thruway reported selling an average of u = 410 packs per day with o = 60. The distribution of daily sales was approximately normal. For a sample of n= 9 days following the new tax, the researcher found an average of M = 386 packs per day for the same stores. a. Is the sample mean sufficient to conclude that there was a significant change in cigarette purchases after the new tax. Use a two-tailed test with a = .05. b. If the population standard deviation was o=30, is the result sufficient to conclude that there is a significant difference? c. Explain why the two tests lead to different outcomes.
ANSWER: a. The standard error is oM = 20 and z=-24/20=-1.20. The boundaries for the critical region lie at -1.96 and +1.96. Therefore, fail to reject HO and conclude that there is no significant change in cigarette purchases. b. With o=30, the standard error is o M = 10 and z=-2.40. Reject HO. c. The smaller variability produces a smaller standard error. As a result, the 24-point difference between the sample mean and the population mean is now significant.
26. If a treatment has a very small effect, then what is a likely outcome for a hypothesis test evaluating the treatment? a. A Type I error b. A Type II error c. Correctly reject the null hypothesis d. Correctly fail to reject the null hypothesis
B
74. A population has a mean of u = 80 with o = 20. a. If a single score is randomly selected from this population, how much distance, on average, should you find between the score and the population mean? b. If a sample of n=4 scores is randomly selected from this population, how much distance, on average, should you find between the sample mean and the population mean? c. If a sample of n= 100 scores is randomly selected from this population, how much distance, on average, should you find between the sample mean and the population mean?
ANSWER: a. o = 20 points b. OM = 10 points C. OM= 2 points
77. For a normal population with u = 100 and o=20: a. What is the probability of obtaining a sample mean greater than 110 for a sample of n= 4 scores? b. What is the probability of obtaining a sample mean greater than 110 for a sample of n= 16 scores? c. For a sample of n=25 scores, what is the probability that the sample mean will be within 5 points of the population mean? In other words, what is p(95<M< 105)?
ANSWER: a. z= 10/10 = 1.00. p=0.1587 b. z= 10/5 = 2.00. p=0.0228 c. z=+5/4 = +1.25. p = 2(0.3944)=0.7888
75. Each of the following samples was obtained from a population with u = 100 and o= 10. Find the Z-score corresponding to each sample mean. a. M=95 for a sample of n=4 b. M = 104 for a sample of n=25 c. M = 103 for a sample of n= 100
ANSWER: a. z=-1.00 b. z= +2.00 c. z=+3.00
19. Which of the following is an accurate definition of a Type I error? a. Rejecting a false null hypothesis b. Rejecting a true null hypothesis c. Failing to reject a false null hypothesis d. Failing to reject a true null hypothesis
B
2. What is measured by the numerator of the z-score test statistic? a. The average distance between M and u that would be expected if Ho was true b. The actual distance between M and u c. The position of the sample mean relative to the critical region d. Whether or not there is a significant difference between M and u
B
26. A sample of n= 4 scores is selected from a population with u = 50 and o= 12. If the sample mean is M = 56, what is the z-score for this sample mean? a. 0.50 b. 1.00 c. 2.00 d. 4.00 ANSWER: b
B
1. Which of the following accurately describes a hypothesis test? a. A descriptive technique that allows researchers to describe a sample b. A descriptive technique that allows researchers to describe a population c. An inferential technique that uses the data from a sample to draw inferences about a population d. An inferential technique that uses information about a population to make predictions about a sample
C
14. Even if a treatment has an effect, it is still possible to obtain a sample mean that is very similar to the original population mean. What outcome is likely if this happens? a. Reject Ho and make a Type I error. b. Correctly reject Ho. c. Fail to reject H, and make a Type II error. d. Correctly fail to reject Ho.
C
16. By selecting a larger alpha level, a researcher is a. attempting to make it more difficult to reject Ho b. less able to detect a treatment effect c. increasing the risk of a Type I error d. decreasing the risk of a Type I error
C
20. Which of the following is an accurate definition of a Type II error? a. Rejecting a false null hypothesis b. Rejecting a true null hypothesis c. Failing to reject a false null hypothesis d. Failing to reject a true null hypothesis
C
25. A researcher administers a treatment to a sample of participants selected from a population with u = 80. If a hypothesis test is used to evaluate the effect of the treatment, which combination of factors is most likely to result in rejecting the null hypothesis? a. A sample mean near 80 with a=.05 b. A sample mean near 80 with a = .01 c. A sample mean much different than 80 with a =.05 d. A sample mean much different than 80 with a =.01
C
28. A sample of n= 16 scores is obtained from a population with u = 50 and o= 16. If the sample mean is M= 54, then what is the z-score for the sample mean? a. z=0.25 b. z=0.50 c. z=1.00 d. z= 4.00 ANSWER: C
C
29. A sample of n=9 scores is obtained from a population with u = 70 and 5= 18. If the sample mean is M = 76, then what is the z-score for the sample mean? a. z=0.33 b. z=0.50 c. z=1.00 d. z=3.00 ANSWER: c
C
30. If a sample of n = 4 scores is obtained from a population with u = 70 and o = 12, then what is the Z-score corresponding to a sample mean of M = 76? a. z=0.25 b. z=0.50 c. z=1.00 d. z=2.00 ANSWER: C
C
31. Which combination of factors will increase the chances of rejecting the null hypothesis? a. A large standard error and a large alpha level b. A large standard error and a small alpha level c. A small standard error and a large alpha level d. A small standard error and a small alpha level
C
36. A random sample of n= 4 scores is obtained from a normal population with u = 20 and o = 4. What is the probability that the sample mean will be greater than M = 22? a. 0.50 b. 1.00 c. 0.1587 d. 0.3085 ANSWER: c
C
39. Which of the following is an accurate definition for the power of a statistical test? a. The probability of rejecting a true null hypothesis b. The probability of supporting true null hypothesis c. The probability of rejecting a false null hypothesis d. The probability of supporting a false null hypothesis
C
40. A sample of n= 16 scores is selected from a population with u = 100 and o = 32. If the sample mean is M = 104, what is the z-score for this sample mean? a. 2.00 b. 1.00 c. 0.50 d. 0.25 ANSWER: c
C
40. Which of the following will increase the power of a statistical test? a. Change a from .05 to .01 b. Change from a one-tailed test to a two-tailed test c. Change the sample size from n=25 to n= 100 d. None of the other three options will increase the power.
C
37. A random sample of n=9 scores is obtained from a normal population with u = 40 and o= 6. What is the probability that the sample mean will be greater than M=43? a. 0.3085 b. 0.6915 c. 0.9332 d. 0.0668 ANSWER: d
D
11. If a is held constant at .05, what is the impact of changing the sample size on the critical region and the risk of a Type I error? a. As sample size increases, the critical region expands and the risk of a Type I error increases. b. As sample size increases, the critical region shrinks and the risk of a Type I error increases. c. As sample size increases, the critical region expands and the risk of a Type I error decreases. d. As the sample size increases, the critical region, and the risk of a Type I error remain unchanged.
D
11. Samples of size n=9 are selected from a population with u = 80 with o=18. What is the expected value of M, the mean of the distribution of sample means? a. 6 b. 18 c. 80/3 d. 80
D
11. What proportion of a normal distribution is located between z=-0.25 and z= +0.25? a. 0.5987 b. 0.4013 c. 0.0987 d. 0.1974
D
12. What proportion of a normal distribution is located between z=-1.50 and z= +1.50? a. 0.9332 b. 0.0668 c. 0.4332 d. 0.8664
D
17. Decreasing the alpha level from a = .05 to a = .01 . a. increases the probability of a Type I error b. increases the size of the critical region c. increases the probability that the sample will fall into the critical region d. decreases the probability of a Type I error
D
21. What is the consequence of a Type I error? a. Concluding that a treatment has an effect when it really does b. Concluding that a treatment has no effect when it really has no effect c. Concluding that a treatment has no effect when it really does d. Concluding that a treatment has an effect when it really has no effect
D
28. A researcher administers a treatment to a sample of participants selected from a population with u = 80. If a hypothesis test is used to evaluate the effect of the treatment, which combination of factors is most likely to result in rejecting the null hypothesis? a. A sample mean near 80 for a small sample b. A sample mean near 80 for a large sample c. A sample mean much different than 80 for a small sample d. A sample mean much different than 80 for a large sample
D
32. A researcher is conducting an experiment to evaluate a treatment that is expected to increase the scores for individuals in a population which is known to have a mean of u = 80. The results will be examined using a one-tailed hypothesis test. Which of the following is the correct statement of the null hypothesis? a. l> 80 b. u > 80 c. u< 80 d. u < 80
D
35. For a normal population with u = 40 and o= 10 which of the following samples has the highest probability of being obtained? a. M = 42 for a sample of n= 4 b. M = 44 for a sample of n = 4 c. M=42 for a sample of n=100 d. M = 44 for a sample of n= 100 ANSWER: d
D
5. A sample of n=25 individuals is selected from a population with u = 80 and a treatment is administered to the sample. What is expected if the treatment has no effect? a. The sample mean should be very different from 80 and should lead you to reject the null hypothesis. b. The sample mean should be very different from 80 and should lead you to fail to reject the null hypothesis. c. The sample mean should be close to 80 and should lead you to reject the null hypothesis. d. The sample mean should be close 80 and should lead you to fail to reject the null hypothesis.
D
43. The null hypothesis states that the sample mean (after treatment) is equal to the original population mean (before treatment). a. True b. False
F
45. A sample of n=25 scores is selected from a population with a mean of u = 80 and a standard deviation of o = 20. The standard error for the sample mean is 20. a. True b. False ANSWER: False
F
46. As the sample size increases, the standard error also increases. a. True b. False ANSWER: False
F
46. If a hypothesis test leads to rejecting the null hypothesis, it means that the data did not provide enough evidence to conclude that the treatment has an effect. a. True b. False
F
47. If the sample data are in the critical region with a = .05, then the same sample data would still be in the critical region if a were changed to .01. a. True b. False
F
47. The mean for a sample of n = 4 scores has a standard error om = 5 points. This sample was selected from a population with a standard deviation of o=20. a. True b. False ANSWER: False
F
49. A Type I error occurs when a treatment actually does have an effect on the scores but the effect was not large enough to reject the null hypothesis. a. True b. False
F
41. All probabilities can be expressed as decimal values ranging from 0 to 1.00. a. True b. False
T
41. The null hypothesis is stated in terms of the population, even though the data come from a sample. a. True b. False
T
41. Two samples probably will have different means even if they are both the same size and they are both selected from the same population. a. True b. False ANSWER: True
T
9. A researcher conducts a hypothesis test to evaluate the effect of a treatment that is expected to increase scores. The hypothesis test produces a z-score of z= 2.37. If the researcher is using a one-tailed test, what is the correct statistical decision? a. Reject the null hypothesis with a = .05 but not with a=.01. b. Reject the null hypothesis with either a = .05 or a=.01. c. Fail to reject the null hypothesis with either a =.05 or a=.01. d. This cannot be answered without additional information. ANSWER: b
a. The population before treatment b. The population after treatment c. The sample before treatment d. The sample after treatment ANSWER: B
1. Which of the following are requirements of a random sample? a. Every individual has an equal chance of being selected. b. The probabilities cannot change during a series of selections. c. There must be sampling with replacement. d. There must be at least 100 observations.
A
10. A vertical line is drawn through a normal distribution at z= 1.20. What proportion of the distribution is on the left-hand side of the line? a. 0.8849 b. 0.1151 c. 0.3849 d. 0.7698
A
14. What is the probability of randomly selecting a z-score less than z=1.25 from a normal distribution. a. 0.8944 b. 0.1056 c. 0.3944 d. 0.2112
A
19. A normal distribution has a mean of u = 40 with o= 10. What proportion of the scores in this distribution are smaller than X = 35? a. 0.3085 b. 0.6915 c. 0.0668 d. 0.9332
A
22. A normal distribution has a mean of u = 70 with o = 12. If one score is randomly selected from this distribution, what is the probability that the score will be greater than X = 58? a. 0.8413 b. 0.1577 c. 0.3413 d. 0.6826
A
23. A sample obtained from a population with o = 12 has a standard error of 2 points. How many scores are in the sample? a. n= 36 b. n=24 c. n=6 d. n=3 ANSWER: a
A
32. Scores on the SAT form a normal distribution with a mean of u = 500 with o= 100. If the state college only accepts students who score in the top 60% on the SAT, what is the minimum score needed to be accepted? a. X=475 b. X= 525 c. X=440 d. X= 560
A
8. A vertical line is drawn through a normal distribution at z=0.50 and separates the distribution into two sections. What the proportion of the distribution is in the larger section? a. 0.6915 b. 0.3085 c. 0.1915 d. 0.3830
A
75. For a normal distribution with u = 200 and o=50, find the following values: a. What X value separates the highest 10% of the distribution from the rest of the scores? b. What X values form the boundaries for the middle 60% of the distribution? c. What is the probability of randomly selecting a score greater than X = 325? N
ANSWER: a. z= 1.28, X = 264 b. z=0.84, X = 158 to 242 c. z= 2.50, p = 0.0062 N
73. A normal distribution has a mean of u = 100 with o=20. Find the following probabilities: a. p(X> 102) b.p(X<65) c.p(X<130) d. p(95<X< 105)
ANSWER: a. z=0.10, p=0.4602 b. z=-1.75, p=0.0401 c. z=1.50, p=0.9332 d. -0.25<z< +0.25, p=0.1974
71. Assume that a vertical line is drawn through a normal distribution at each of the following z-score locations. In each case, determine whether the tail is on the left side or the right side of the line and find the proportion of the distribution that is located in the tail. a. z= +1.80 b. z= +0.60 c. Z=-0.40 d. z=-1.25
ANSWER: a. The tail is on the right. p=0.0359 b. The tail is on the right. p=0.2743 c. The tail is on the left. p=0.3446 d. The tail is on the left. p=0.1056
76. In an ESP experiment subjects must predict whether a number randomly generated by a computer will be odd or even. a. What is the probability that a subject would guess exactly 18 correct in a series of 36 trials? b. What is the probability that a subject would guess more than 20 correct in a series of 36 trials?
ANSWER: a. With n= 36 and p=q= 1/2, you may use the normal approximation with u = 18 and o=3. X= 18 has real limits of 17.5 and 18.5 corresponding to z=0.17 and z=+0.17.p=0.1350. b. p(X> 20.5) =p(z>0.83)
72. For a normal distribution, a. What z-score separates the highest 10% from the rest of the scores? b. What z-score separates the highest 30% from the rest of the scores? c. What z-score separates the lowest 40% from the rest of the scores? d. What z-score separates the lowest 20% from the rest of the scores?
ANSWER: a. z= 1.28 b. z=0.52 c. Z=-0.25 d. z=-0.84
74. Assume that the total score (from both teams) for college football games averages u = 42 points per game and that the distribution of total points is approximately normal with o=20. a. What is the probability that a randomly selected game would have more than 60 points? b. What proportion of college football games have a point total between 20 and 60?
ANSWER: a. z=0.90, p=0.1841 b. -1.10 <z<0.90, p=0.6802
13. What is the probability of randomly selecting a z-score greater than z=0.75 from a normal distribution? a. 0.7734 b. 0.2266 c. 0.2734 d. 0.4532
B
16. If random samples, each with n= 36 scores, are selected from a normal population with u = 80 and o= 18, how much difference, on average, should there be between a sample mean and the population mean? a. 2 points b. 3 points c. 6 points d. 18 points
B
18. What happens to the standard error of M as sample size increases? a. It also increases. b. It decreases. c. It stays constant. d. The standard error does not change in a predictable manner when sample size increases.
B
2. A jar contains 10 red marbles and 30 blue marbles. What is the probability of randomly selecting a red marble from the jar? a. 10/30 b. 10/40 c. 1/10 d. 1/40
B
2. What term is used to identify the standard deviation of the distribution of sample means? a. the expected value of M b. the standard error of M c. the sample mean d. the central limit mean
B
20. A normal distribution has u = 80 and o= 10. What is the probability of randomly selecting a score greater than 90 from this distribution? a. p=0.8413 b. p=0.1587 c. p=0.3085 d. p=0.6915
B
20. For a particular population, a sample of n=9 scores has a standard error of 8. For the same population, a sample of n = 16 scores would have a standard error of_ . a. 8 b. 6 c. 4
B
21. A normal distribution has a mean of u = 70 with o = 12. If one score is randomly selected from this distribution, what is the probability that the score will be greater than X = 79? a. 0.7734 b. 0.2266 c. 0.2734 d. 0.3085
B
22. A sample of n=9 scores has a standard error of 6. What is the standard deviation of the population from which the sample was obtained? a. 54 b. 18 c. 6 d. 2.
B
10. A random sample of n= 36 scores is selected from a population. Which of the following distributions will definitively be normal? a. The scores in the sample will form a normal distribution. b. The scores in the population will form a normal distribution. c. The distribution of sample means will form a normal distribution. d. None of the distributions - the sample, the population, the distribution of sample means - will definitely be normal.
C
12. Samples of size n=9 are selected from a population with u = 80 with o = 18. What is the standard error for the distribution of sample means? a. 80 b. 18 c. 6 d. 2
C
14. If random samples, each with n=4 scores, are selected from a normal population with u = 80 and 6 = 36, then what is the standard error for the distribution of sample means? a. 4 b. 9 c. 18 d. 36
C
15. If all the possible random samples with n = 36 scores are selected from a normal population with u = 80 and o=18, and the mean is calculated for each sample, then what is the average of all the sample means? a. 2 b. 6 c. 80 d. It cannot be determined without additional information.
C
16. What z-score value separates the top 70% of a normal distribution from the bottom 30%? a. z=0.52 b. z=0.84 c. Z=-0.52 d. z=-0.84
C
17. What happens to the expected value of M as sample size increases? a. It also increases. b. It decreases. c. It stays constant. d. The expected value does not change in a predictable manner when sample size increases.
C
18. A normal distribution has a mean of u = 40 with o=10. What proportion of the scores in this distribution is greater than X= 55? a. 0.3085 b. 0.6915 c. 0.0668 d. 0.9332
C
19. Which combination of factors will produce the smallest value for the standard error? a. A large sample and a large standard deviation b. A small sample and a large standard deviation c. A large sample and a small standard deviation d. A small sample and a small standard deviation
C
25. A normal distribution has a mean of u = 100 with o = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 80 and X = 100? a. 0.8413 b. 0.1587 c. 0.3413 d. 0.6826
C
27. A normal distribution has a mean of u = 100 with o = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 90 and X = 120? a. 0.1498 b. 0.4672 c. 0.5328 d. 0.2996
C
35. Under what circumstances does the binomial distribution approximate a normal distribution? a. when pn > 10 b. when qn > 10 c. when pn and qn are both > 10 d. when npq10
C
39. A true/false test has 100 questions. Using the normal approximation to the binomial distribution, what is the probability of getting 55 or more correct by just guessing? a. p(X> 55) b. p(X> 55.5) c. p(X > 54.5) d. p(X> 45)
C
5. A sample of n= 100 scores is selected from a population with u = 80 with o= 20. On average, how much error is expected between the sample mean and the population mean? a. 0.2 points b. 0.8 points c. 2 points d. 4 points
C
6. What proportion of a normal distribution is located between the mean and z= 1.40? a. 0.9192 b. 0.0808 c. 0.4192 d. 0.8384
C
7. What proportion of a normal distribution is located between the mean and z=-0.40? a. 0.6554 b. 0.3446 c. 0.1554 d. 0.3108
C
7. What symbol is used to identify the standard error of M? a. OM b. u c. σ/ d. MM
C
13. If random samples, each with n=9 scores, are selected from a normal population with u = 80 and o= 36, then what is the expected value of the mean of the distribution of sample means? a. 4 b. 12 c. 16 d. 80
D
17. What z-score values form the boundaries for the middle 60% of a normal distribution? a. z=+0.25 and z=-0.25 b. z=+0.39 and z=-0.39 c. z=+0.52 and z=-0.52 d. z= +0.84 and z=-0.84
D
26. A normal distribution has a mean of u = 100 with o = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 90 and X = 110? a. 0.6915 b. 0.3085 c. 0.1915 d. 0.3830
D
33. John drives to work each morning and the trip takes an average of u = 38 minutes. The distribution of driving times is approximately normal with a standard deviation of o= 5 minutes. For a randomly selected morning, what is the probability that John's drive to work will take less than 35 minutes? a. 0.6554 b. 0.3446 c. 0.7257 d. 0.2743
D
34. John drives to work each morning and the trip takes an average of u = 38 minutes. The distribution of driving times is approximately normal with a standard deviation of o= 5 minutes. For a randomly selected morning, what is the probability that John's drive to work will take between 36 and 40 minutes? a. 0.0793 b. 0.1526 c. 0.1554 d. 0.3108
D
36. For a binomial distribution with p=q=1/2 and n= 36, what is the mean for the normal approximation? a. 3 b. 9 c. 16 d. 18
D
4. The distribution of sample means a. is always a normal distribution b. will be normal only if the population distribution is normal c. will be normal only if the sample size is at least n= 30 d. will be normal if either the population is normal or the sample size is n > 30
D
8. Under what circumstances is the distribution of the sample means normal? a. It is always normal b. Only if the population distribution is normal c. If the sample size is greater than 30 d. If the population is normal or if the sample size is greater than 30
D
9. A random sample of n=4 scores is selected from a population. Which of the following distributions will definitively be normal? a. The scores in the sample will form a normal distribution. b. The scores in the population will form a normal distribution. c. The distribution of sample means will form a normal distribution. d. None of the distributions - the sample, the population, the distribution of sample means -will definitely be normal.
D
42. A jar contains 10 red marbles and 20 blue marbles. If you take a random sample with the replacement of two marbles from this jar and the first marble is blue, then the probability that the second marble is blue is p= 19/29. a. True b. False
F
43. For a normal distribution, proportions in the right-hand tail are positive and proportions in the left-hand tail are negative. a. True b. False
F
45. A vertical line drawn through a normal distribution at z=-0.75 will separate the distribution into two sections. The proportion in the smaller section is 0.2734. a. True b. False
F
56. If one score is randomly selected from a normal distribution with u = 100 and o=20, the probability of obtaining a score greater than X = 110 is p = 0.6915. a. True b. False
F
62. If one score is randomly selected from a normal distribution with u = 100 and o=20, the probability of obtaining a score between X = 90 and X = 100 is p=0.3085. a. True b. False
F
64. A binomial distribution with p=2/3 and n= 24 meets the criterion for using the normal approximation. a. True b. False
F
66. The binomial distribution for p= 1/2 and n= 100 has a standard deviation of o = 25. a. True b. False
F
55. For a normal distribution, the Z-score boundary that separates the lowest 2.5% of the scores from the rest is z= 1.96. a. True b. False
T
59. For a normal distribution with u = 80 and o= 10 the score that separates the bottom 10% of the distribution from the rest is 67.2. a. True b. False
T
60. If one score is randomly selected from a normal distribution with u = 100 and o= 20, the probability of obtaining a score less than X = 95 is p=0.4013. a. True b. False
T
67. For a binomial distribution, the probability of obtaining a score greater than 19 is computed as p(X> 19.5). a. True b. False
T
68. For a binomial distribution, the probability of obtaining a score of X = 19 or greater is computed as p(X> 18.5). a. True b. False
T
69. For the normal approximation to the binomial distribution with n= 100 and p= 1/2; a score of X = 60 corresponds to a Z-score of z=2.00. a. True b. False
T
