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essentials of statistics

Essentials Of Statistics For The Behavioral Sciences 5th Edition Frederick J Gravetter, Larry B. Wallnau - Solutions

27. A population consists of the following N = 6 scores:0,4,6, 1,3, and 4.a. Compute u. and a for the population.b. Find the z-score for each score in the population.c. Transform the original population into a new population of N = 6 scores with a mean of u. = 50 and a standard deviation of a = 10.
26. A distribution with a mean of p = 86 and a standard deviation of cr = 12 is being transformed into a standardized distribution with p = 100 and ct = 20. Find the new, standardized score for each of the following values from the original population.a. X - 80b. X = 89c. X = 95d. X = 98
25. A distribution with a mean of p = 38 and a standard deviation of ct = 4 is being transformed into a standardized distribution with p = 50 and ct = 10. Find the new, standardized score for each of the following values from the original population.a. X = 42b. X = 40c. X = 38d. X = 36
24. Suppose that you got a score of X = 78 on an English test for which the mean was p = 70 and the standard deviation was ct = 10. Also, suppose that you got a score of X = 64 on a Spanish test with p = 50 and ct = 7. For which test would you expect the better grade? Explain your answer.
23. On Tuesday afternoon, Bill earned a score of X = 73 on an English test with p = 65 and ct = 8. The same day, John earned a score of X = 63 on a math test with p = 57 and ct = 3. Who should expect the better grade, Bill or John? Explain your answer.
22. Answer the question as in Problem 21, but this time assume that the mean for the exam is p = 60.
21. Suppose that you have a score of X = 55 on an exam with p, = 50. Which standard deviation would give you the better grade: a = 5 or ct = 10?
20. For each of the following populations, would a score of X = 48 be considered a central score (near the middle of the distribution) or an extreme score (far out in the tail of the distribution)?a. p = 40 and ct = 10b. p = 40 and cr = 2c. p = 50 and a = 4d. p = 60 and cr = 4
19. In a distribution of scores, a raw score of X = 43 corresponds to z = 1 .00 and a score of X = 49 corresponds to z = 2.00. Find the mean and the standard deviation for the distribution of scores.
18. For a population of exam scores, a score of X = 58 corresponds to z = +0.50 and a score of X = 46 corresponds to z = - 1.00. Find the mean and the standard deviation for the population. (Hint: Sketch the distribution and locate the two scores in your sketch.)
17. For a population with a mean of p = 70, a score of 62 corresponds to a z-score of z = —2.00. What is the population standard deviation?
16. For a population with a mean of p. = 85, a score of 90 corresponds to a z-score of z = 1 .00. What is the population standard deviation?
15. For a population with a standard deviation of ct = 10, a score of X = 65 corresponds to z = 1 .50. What is the population mean?
14. For a population with a standard deviation of a = 4, a score of X = 44 corresponds to z = -0.50. What is the population mean?
13. A score that is 8 points above the mean corresponds to a z-score of z = 0.50. What is the population standard deviation?
12. A score that is 10 points below the mean corresponds to a z-score of z = —2.00. What is the population standard deviation?
11. Find the X value corresponding to z = +1.50 for each of the following distributions.a. p = 100 and a = 10b. p = 100 and a = 20c. p = 80 and cr = 4d. p = 80 and a = 2
10. Find the z-score corresponding to a score of X = 50 for each of the following distributions.a. p. = 60 and a = 5b. p = 40 and a = 5c. p = 60 and
9. A population of scores has p = 85 and ct = 20. Find the raw score (X value) corresponding to each of the following z-scores in this population:z = 1.25 z = 0.60= -2.30= -0.40-1.50 2.10
8. A population of scores has p. = 60 and ct = 12. Find the z-score corresponding to each of the following X values from this population:X = 66 X = 48 X = 84 X = 55 X = 70 X = 75
7. A population has a mean of u, = 70 and a standard deviation of a = 8.a. For this population, find the z-score corresponding to each of the following scores.X = 74 X = 62 X = 68 X= 82 X = 86 X = 54b. For the same population, find the score (X value)corresponding to each of the following
6. A population has a mean of p. = 50 and a standard deviation of a = 10.a. For this population, find the z-score corresponding to each of the following scores.X = 55 X = 48 X = 40 X= 70 X = 35 X = 65b. For the same population, find the score (X value)corresponding to each of the following
5. For a population with u, = 45 and a = 7, find the z-score for each of the following X values. (Note: You probably will need to use the formula and a calculator to find these values.)X = 47 X = 35 X = 40 X = 60 X = 55 X = 42
4. For a population with p. = 80 and a = 20,a. Find the z-score for each of the following X values.(Note: You should be able to find these values using the definition of a z-score. You should not need to use a formula or do any serious calculations.)X = 75 X = 95 X = 90 60 110 40b. Find the score
3. A distribution has a standard deviation of o = 10. For each of the following z-scores, determine whether the location is above or below the mean and determine how many points away from the mean. For example.z = + 1 .00 corresponds to a location that is above the mean by 10 points.a. z = +2.00b.
2. A distribution has a standard deviation of a — 4. Find the z-score for each of the following locations in the distribution.a. Above the mean by 4 pointsb. Above the mean by 1 2 pointsc. Below the mean by 2 pointsd. Below the mean by 8 points
1. Describe exactly what information is provided by a z-score.
26. When the sun cannot be seen (overcast day), homing pigeons find their way back to their roosts by using magnetic cues from the Earth (Walcott, 1972).Consider the following study. One sample of pigeons has a magnet fastened on their heads to interfere with their ability to detect the Earth's
25. A study examines the relationship between level of arousal and problem solving. Three samples are used, consisting of participants with low, moderate, or high levels of arousal. The researcher measures the number of problems successfully completed during a problemsolving task. The data are as
24. For the following population of scores:1, 6, 9, 0, 4a. Find the mean for the population, and compute the deviation score for each individual.b. Show that the deviation scores sum to zero.c. Square each deviation, and find the sum of squared deviations (SS).d. Now assume that the set of scores
23. For the following population of N = 4 scores:2, 0, 8, 2a. Use the definitional formula to compute SS; then find the population variance and the standard deviation.Add 3 points to each score; then compute SS, variance, and standard deviation for the new population.c. Multiply each of the
22. Sketch a normal distribution (see Figure 4.5, page 92)with p. = 50 and a = 20.a. Locate each of the following scores in your sketch, and indicate whether you consider each score to be an extreme value (high or low) or a central value:65, 55, 40, 47b. Make another sketch showing a distribution
21. For the following population of N = 12 scores:6, 10, 4, 4, 6, 7, 11. 7, 3, 11, t 12a. Sketch a histogram showing the population distribution.b. Locate the value of the population mean in your sketch, and make an estimate of the standard deviation(as done in Example 4.2).c. Compute SS, variance,
20. For the following population of N = 5 scores:11, 2, 0, 8, 4a. Sketch a histogram showing the population distribution.b. Locate the value of the population mean in your sketch, and make an estimate of the standard deviation(as done in Example 4.2).c. Compute SS, variance, and standard deviation
19.a. Sketch a histogram showing the sample distribution.b. Locate the value of the sample mean in your sketch, and make an estimate of the sample standard deviation (as done in Example 4.5).c. Compute SS, variance, and standard deviation for the sample. (How well does your estimate compare with
18. For the following sample of n = 5 scores:10. 0. 6. 2, 2
17. Calculate 55, variance, and standard deviation for the following population of N = 4 scores: 6, 8, 0, 6.(Note: The definitional formula for 55 works well with these scores.)
16. Calculate 55, variance, and standard deviation for the following population of N = 6 scores: 5, 0, 9, 3, 8, 5.(Note: The definitional formula for 55 works well with these scores.)
15. Calculate 55, variance, and standard deviation for the following sample of n = 9 scores: 2, 0, 0, 0, 0, 2, 0, 2, 0. (Note: The computational formula for 55 works best with these scores.)
14. Calculate 55, variance, and standard deviation for the following sample of n = 4 scores: 0, 3, 0, 3. (Note:The computational formula for 55 works best with these scores).
13. There are two different formulas or methods that can be used to calculate 55.a. Under what circumstances is the definitional formula easy to use?b. Under what circumstances is the computational formula preferred?
12. For the following scores:1, 0, 4, 1a. Calculate the mean. (Note that the value of the mean does not depend on whether the set of scores is considered to be a sample or a population.)b. Find the deviation for each score, and check that the deviations sum to zero.c. Square each deviation, and
11. Calculate 55, variance, and standard deviation for the following sample:4, 7, 3, 1. 5
10.a. A sample of n = 6 scores has 55 = 60. What is the variance for this sample?b. A population of N = 6 scores has 55 = 60. What is the variance for this population?
9. On an exam with |jl = 75, you obtain a score of X = 80.a. Would you prefer that the exam distribution had a = 2 or
8. A distribution of scores has a mean of M = 42.a. If the standard deviation is s = 12, would a score of X = 48 be considered an extreme value? Explain your answer.b. If the standard deviation is s — 2, would a score of X — 48 be considered an extreme value? Explain your answer.
7. For the data in the following sample:1, 4, 3, 6, 2, 7, 18, 3, 7, 2, 4, 3a. Sketch a frequency distribution histogram.b. Compute the mean and standard deviation.c. Find the median and the semi-interquartile range.d. Which measures of central tendency and variability provide a better description
6. For the following sample:2, 2, 4, 1, 3, 2, 1, 2a. Calculate the range, the interquartile range, and the standard deviation.b. Add two points to every score, then compute the range, the interquartile range, and the standard deviation again. How is variability affected by adding a constant to
5. Explain what it means to say that the sample variance provides an unbiased estimate of the population variance.
4. In general, what does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample.
3. Can 55 ever have a value less than zero? Explain your answer.
2. A population has (jl = 100 and a = 20. If you select a single score from this population, on the average, how close would it be to the population mean? Explain your answer.
1. In words, explain what is measured by each of the following:a. 55b. variancec. standard deviation
26. Does it ever seem to you that the weather is nice during the work week, but lousy on the weekend'7 Cerveny and Balling (1988) have confirmed that this is not your imagination—pollution accumulating during the work week most likely spoils the weekend weather for people on the Atlantic coast.
25.In a classic study investigating heredity and intelligence, a large sample of rats was tested on a maze(Tryon, 1940). Based on their error scores, the brightest and the dullest rats were selected from the sample.The brightest males and females were mated to produce a strain of "maze-bright"
24. On a standardized reading achievement test, the nationwide average for seventh-grade children is jjl =7.0. A seventh-grade teacher is interested in comparing class reading scores with the national average. The scores for the 1 6 students in this class are as follows:8, 6. 5, 10, 5. 6, 8. 9, 7.
23. The following frequency distribution summarizes the number of absences for each student in a class of n = 20:Number of absences (X) /5 or more 3 4 4 3 3 2 6 1 3 1a. Find the mode for this distribution.b. Find the median number of absences for this class.c. Explain why you cannot compute the
22. For each of the following situations, identify the measure of central tendency (mean, median, or mode) that would provide the best description of the "average"score:a. A researcher asks each individual in a sample of 50 adults to name his/her favorite season (summer, fall, winter, spring).b. An
21. A sample of n = 20 scores has a mean of M = 40. A«• second sample has n = 5 scores with a mean of M —30. If the two samples are combined, what value will be obtained for the mean of the combined sample?
20. A sample of n = 4 scores has a mean of M = 12. A second sample of n = 6 scores has a mean of M = 8.If the two samples are combined, what value will be obtained for the mean of the combined sample?
19. One sample has a mean of M = 4 and a second sample has a mean of M = 8. The two samples are combined into a single set of scores.a. What is the mean for the combined set if both of the original samples have n = 1 scores?b. What is the mean for the combined set if the first sample has n = 3 and
18. A sample of n = 8 scores has a mean of M = 12. One new score is added to the sample and the new mean is found to be M = 13. What is the value of the new score?
17. A sample of n 1 scores has a mean of M = 5. One score is removed from the sample and the mean for the remaining scores is found to he M = 4. What is the value of the score that was removed? (Hint: Find IX for the original scores and lor the new sample.)PROBLEMS 79
16. A population of N = 10 scores has a mean of u, = 9.If one score with a value of X = 9 is removed from the population, what will be the new value for the population mean '
15. A sample of n = 6 scores has a mean of M = 30. If one new score with a value of X = 16 is added to the sample, what will be the new value for the sample mean?
14. A set of seven scores has a mean of 10. If one of the scores is changed from X = 15 to X = 1, what will be the value for the new mean?
13. A sample of n = 8 scores has a mean of M = 20. If one of the scores is changed from X = 4 to X = 12, what will be the new value for the sample mean?
12. A sample of n = 12 scores has a mean of M = 6.What is the value of XX for this sample?
11. A population of N = 7 scores has a mean of p. = 9.What is the value of XX for this population?
10. Find the mean, median, and mode for the set of scores in the following frequency distribution table:X /5 1 4 2 3 3 21 51 X /10 1 9 9 8 5 7 3 6 1 5 1
9. Find the mean, median, and mode for the set of scores in the following frequency distribution table:
8. Find the mean, median, and mode for the following sample of scores:8, 7, 9, 9. 10, 6, 9, 9. 4, 8
7. Find the mean, median, and mode for the following sample of scores:3, 2, 3, 5, 4, 1, 4, 3, 2, 3
6. Explain why the mean is often not a good measure of central tendency for a skewed distribution.
5. Under what circumstances is the mode the preferred measure of central tendency?
4. Under what circumstances will the mean, the median, and the mode all have the same value?
3. Identify the circumstances in which the median rather than the mean is the preferred measure of central tendency.
2. Explain what is meant by each of the following statements:a. The mean is the balance point of the distribution.b. The median is the midpoint of the distribution.
1. Explain the general purpose for measuring central tendency.
25. The following data are quiz scores from two different sections of an introductory statistics course:Section I Section II 9 6 8 4 7 8 10 8 r 6 3 7 7 8 8 4 6 3-7 5 10 10 3 6 9 6 7 7 4 6a. Organize the scores from each section in a frequency distribution histogram.b. Describe the general
Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, participants were shown a list of sentences, of which half were humorous and half were nonhumorous.Schmidt then measured the number of each type of sentence recalled by each participant.
23. Mental imagery has been shown to be very effective for improving memory. In one demonstration study, two groups of participants were presented with a list of 20 words. One group was given instructions to use mental images to help them remember the words, and the second group was given no
22. A psychologist would like to examine the effects of diet on intelligence. Two groups of rats are selected, with 12 rats in each group. One group is fed the regular diet of Rat Chow, whereas the second group has special vitamins and minerals added to their food.After 6 months, each rat is tested
21. Construct a grouped frequency distribution table to organize the following set of scores:206, 350, 590, 473, 450, 483, 112, 380, 584, 620, 743, 816, 685, 592, 712, 727, 686. 592, 542, 490, 684, 491, 520, 380
20. For the following set of quiz scores:3, 5, 4, 6, 2, 3, 4, 1, 4, 3, 7, 7, 3, 4, 5, 8, 2, 4, 7, 10a. Construct a frequency distribution table to organize the scores.b. Draw a frequency distribution histogram for these data.
19. Three sets of data are described by identifying the lowest score and the highest score for each set.Describe how a grouped frequency distribution table should be constructed for each set. That is. give the interval width that you would suggest, and make a list of all class intervals.a. 80-127b.
18. For the following scores, construct a frequency distribution table usinga. An interval width of 5.b. An interval width of 10.64. 75, 50, 67. 86. 66. 62, 64, 71. 47, 57. 74, 63. 67. 56, 65, 70, 87, 48. 50.41. 66, 73. 60. 63. 45, 78, 68. 53. 75
17. For the following set of scores:2, 6, 3, 7, 11, 3, 4, 2, 5, 7, 4, 2, 5. 6. 2, 3, 4, 9, 4, 5, 3, 4, 6, 4, 3a. Construct a frequency distribution table.b. Sketch a histogram for these data.c. Sketch a polygon for these data.d. What is the shape of the distribution?
16. For the following set of scores:5, 6, 2. 3, 6, 5, 6, 4, 1, 5, 6, 3, 4a. Construct a frequency distribution table.b. Sketch a polygon showing the distribution.c. Describe the distribution using the following characteristics:(1) What is the shape of the distribution?(2) What score best identifies
15. Find each of the following values for the set of scores shown in the frequency distribution graph.a. Nb. SX 12 3 4 5 6 7 Scores
14. Find each of the following values for the set of scores summarized in the table:a. Nb. IXc. XX2 X /5 1 4 2 3 4 21 31
13. Find each of the following values for the set of scores shown in the frequency distribution table.a. Nb. £X X /5 2 4 3 3 5 21 1
12. For the set of scores shown in the following frequency distribution table,a. How many scores are in the distribution? (N = ?)b. Find SX for this set of scores.X /4 2 3 5 2 3 1 1
11. Place the following set of scores in a frequency distribution table:1. 3. 1, 1, 4. 1. 4, 5, 6, 2, 1. 1, 5, 1, 3. 2. 1, 6, 2. 4. 5, 2, 3, 2, 3 Compute the proportion and the percentage of individuals with each score. From your frequency distribution table, you should be able to identify the
10. An instructor at the state college recorded the home state for each student in an introductory psychology class and obtained the following results:NY NY PA NY NY MA PA NY NH NY NY VA NY NY NY NY NY OH NY PA OH NY NY NY MA ME NJ NY NY NYa. Place the scores in a frequency distribution table.b.
The following table shows the frequency distribution of birth-order position for a sample of n = 24 students.a. What kind of graph would be appropriate for showing this distribution?b. Sketch the frequency distribution graph.Birth-order position /1 st born 12 2nd born 5 3rd born 6 4th born 5th born
8. An instructor obtained the following set of scores from a 10-point quiz for a class of 26 students:8, 5, 4, 5, 5, 7, 6, 4, 3, 4. 5. 6, 6, 4, 5, 5, 10, 6, 9, 5, I, 2, 6, 7, 4a. Place the scores in a frequency distribution table.b. Sketch a histogram showing the distribution.c. Using your graph,
7. The following are reading comprehension scores for a third-grade class of 18 students:6, 3, 5 5, 5, 6, 6, 5, 4, 4, 3, 6.a. Place the scores in a frequency distribution table.b. Sketch a histogram showing the distribution.c. Using your graph, answer the following questions:(1) What is the shape
6. Sketch a histogram and a polygon showing the distribution of scores presented in the following table:X /5 3 4 7 3 5 2 3 1 1
5. Under what circumstances should you use a grouped frequency distribution instead of a regular frequency distribution?

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