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essentials of statistics
Essentials Of Statistics For The Behavioral Sciences 7th Edition Frederick J. Gravetter, Larry B. Wallnau - Solutions
19. The machinery at a food-packing plant is able to put exactly 12 ounces of juice in every bottle. However, some items such as apples come in variable sizes so it is almost impossible to get exactly 3 pounds of apples in a bag labeled “3 lbs.” Therefore, the machinery is set to put an average
18. The population of SAT scores forms a normal distribution with a mean of 500 and a standard deviation of 100. If the average SAT score is calculated for a sample of n 25 students,a. What is the probability that the sample mean will be greater than M 510? In symbols, what is p(M 510)?b.
17. A population of scores forms a normal distribution with a mean of 80 and a standard deviation of 10.a. What proportion of the scores have values between 75 and 85?b. For samples of n 4, what proportion of the samples will have means between 75 and 85?c. For samples of n 16, what proportion
16. A population of scores forms a normal distribution with a mean of 40 and a standard deviation of 12.a. What is the probability of randomly selecting a score less than X 34?b. What is the probability of selecting a sample of n 9 scores with a mean less than M 34?c. What is the probability of
15. A population of scores forms a normal distribution with a mean of 75 and a standard deviation of 20.a. What proportion of the scores in the population have values less than X 80?b. If samples of n 4 are selected from the population, what proportion of the samples will have means less than M
14. The population of IQ scores forms a normal distribution with a mean of 100 and a standard deviation of 15. What is the probability of obtaining a sample mean greater than M 105,a. for a random sample of n 9 people?b. for a random sample of n 36 people?
13. A random sample is obtained from a normal population with a mean of 30 and a standard deviation of 8. The sample mean is M 33.a. Is this a fairly typical sample mean or an extreme value for a sample of n 4 scores?b. Is this a fairly typical sample mean or an extreme value for a sample of n
12. A population forms a normal distribution with a mean of 80 and a standard deviation of 15. For each of the following samples, compute the z-score for the sample mean and determine whether the sample mean is a typical, representative value or an extreme value for a sample of this size.a. M
11. A sample of n 25 scores has a mean of M 84.Find the z-score for this sample:a. If it was obtained from a population with 80 and 10.b. If it was obtained from a population with 80 and 20.c. If it was obtained from a population with 80 and 40.
10. For a population with a mean of 60 and a standard deviation of 24, find the z-score corresponding to each of the following samples.a. M 63 for a sample of n 16 scoresb. M 63 for a sample of n 36 scoresc. M 63 for a sample of n 64 scores
9. For a sample of n 16 scores, what is the value of the population standard deviation () necessary to have a standard error ofa. M 10 points?b. M 5 points?c. M 2 points?
8. If the population standard deviation is 8, how large a sample is necessary to have a standard error that isa. less than 4 points?b. less than 2 points?c. less than 1 point?
7. For a population with a standard deviation of 10, how large a sample is necessary to have a standard error that is:a. less than or equal to 5 points?b. less than or equal to 2 points?c. less than or equal to 1 point?
6. For a population with a mean of 50 and a standard deviation of 10, how much error, on average, would you expect between the sample mean(M) and the population mean for:a. a sample of n 4 scoresb. a sample of n 16 scoresc. a sample of n 25 scores
5. A population has a standard deviation of 30.a. On average, how much difference should exist between the population mean and the sample mean for n 4 scores randomly selected from the population?b. On average, how much difference should exist for a sample of n 25 scores?c. On average, how much
4. The distribution of sample means is not always a normal distribution. Under what circumstances will the distribution of sample means not be normal?
3. A sample is selected from a population with a mean of 80 and a standard deviation of 20.a. What is the expected value of M and the standard error of M for a sample of n 4 scores?b. What is the expected value of M and the standard error of M for a sample of n 16 scores?
2. Describe the distribution of sample means (shape, expected value, and standard error) for samples of n 36 selected from a population with a mean of 100 and a standard deviation of 12.
1. Briefly define each of the following:a. Distribution of sample meansb. Expected value of Mc. Standard error of M
22. Rochester, New York, averages 21.9 inches of snow for the month of December. The distribution of snowfall amounts is approximately normal with a standard deviation of 6.5 inches. This year, a local jewelry store is advertising a refund of 50% off all purchases made in December, if we finish
21. Laboratory rats commit an average of 40 errors before they solve a standardized maze problem. The distribution of error scores is approximately normal with a standard deviation of 8. A researcher is testing the effect of a new dietary supplement on intelligence. A newborn rat is selected
20. Over the past 10 years, the local school district has measured physical fitness for all high school freshmen.During that time, the average score on a treadmill endurance task has been 19.8 minutes with a standard deviation of 7.2 minutes. Assuming that the distribution is approximately
19. A consumer survey indicates that the average household spends $155 on groceries each week.The distribution of spending amounts is approximately normal with a standard deviation of $25. Based on this distribution,a. What proportion of the population spends more than $175 per week on
18. Information from the Department of Motor Vehicles indicates that the average age of licensed drivers is 39.7 years with a standard deviation of 12.5 years. Assuming that the distribution of drivers’ ages is approximately normal,a. What proportion of licensed drivers are more than 50 years
17. A recent newspaper article reported the results of a survey of well-educated suburban parents. The responses to one question indicated that by age 2, children were watching an average of 60 minutes of television each day. Assuming that the distribution of television-watching times is normal
16. The distribution of SAT scores is normal with 500 and 100.a. What SAT score, X value, separates the top 15% of the distribution from the rest?b. What SAT score, X value, separates the top 20% of the distribution from the rest?c. What SAT score, X value, separates the top 25% of the
15. The distribution of scores on the SAT is approximately normal with a mean of 500 and a standard deviation of 100. For the population of students who have taken the SAT,a. What proportion have SAT scores greater than 700?b. What proportion have SAT scores greater than 550?c. What is the
14. IQ test scores are standardized to produce a normal distribution with a mean of 100 and a standard deviation of 15. Find the proportion of the population in each of the following IQ categories.a. Genius or near genius: IQ over 140b. Very superior intelligence: IQ from 120 to 140c. Average
13. A normal distribution has a mean of 50 and a standard deviation of 12. For each of the following scores, indicate whether the tail is to the right or left of the score and find the proportion of the distribution located in the tail.a. X 53b. X 44c. X 68d. X 38
12. For a normal distribution with a mean of 80 and a standard deviation of 20, find the proportion of the population corresponding to each of the following scores.a. Scores greater than 85.b. Scores less than 100.c. Scores between 70 and 90.
11. Find the z-score boundaries that separate a normal distribution as described in each of the following.a. The middle 20% from the 80% in the tails.b. The middle 50% from the 50% in the tails.c. The middle 95% from the 5% in the tails.d. The middle 99% from the 1% in the tails.
10. Find the z-score location of a vertical line that separates a normal distribution as described in each of the following.a. 20% in the tail on the leftb. 40% in the tail on the rightc. 75% in the body on the leftd. 99% in the body on the right
9. Find each of the following probabilities for a normal distribution.a. p(–0.25 z 0.25)b. p(–2.00 z 2.00)c. p(–0.30 z 1.00)d. p(–1.25 z 0.25)
8. What proportion of a normal distribution is located between each of the following z-score boundaries?a. z 0.50 and z 0.50b. z 0.90 and z 0.90c. z 1.50 and z 1.50
7. Find each of the following probabilities for a normal distribution.a. p(z 0.25)b. p(z 0.75)c. p(z 1.20)d. p(z 1.20)
6. Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the body is on the right or left side of the line and find the proportion in the body.a. z 2.20b. z 1.60c. z 1.50d. z 0.70
5. Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the tail is on the right or left side of the line and find the proportion in the tail.a. z 2.00b. z 0.60c. z 1.30d. z 0.30
4. What is sampling with replacement, and why is it used?
3. What are the two requirements that must be satisfied for a random sample?
2. A kindergarten class consists of 14 boys and 11 girls.If the teacher selects children from the class using random sampling,a. What is the probability that the first child selected will be a girl?b. If the teacher selects a random sample of n 3 children and the first two children are both boys,
1. A local hardware store has a “Savings Wheel” at the checkout. Customers get to spin the wheel and, when the wheel stops, a pointer indicates how much they will save. The wheel can stop in any one of 50 sections. Of the sections, 10 produce 0% off, 20 sections are for 10% off, 10 sections for
26. A sample consists of the following n 6 scores: 2, 7, 4, 6, 4, and 7.a. Compute the mean and standard deviation for the sample.b. Find the z-score for each score in the sample.c. Transform the original sample into a new sample with a mean of M 50 and s 10.
25. A population consists of the following N 5 scores:0, 6, 4, 3, and 12.a. Compute and 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 5 scores with a mean of 60 and a standard deviation of 8.
24. A distribution with a mean of 56 and a standard deviation of 20 is being transformed into a standardized distribution with 50 and 10.Find the new, standardized score for each of the following values from the original population.a. X 46b. X 76c. X 40d. X 80
23. A distribution with a mean of 62 and a standard deviation of 8 is being transformed into a standardized distribution with 100 and 20.Find the new, standardized score for each of the following values from the original population.a. X 60b. X 54c. X 72d. X 66
22. For each of the following, identify the exam score that should lead to the better grade. In each case, explain your answer.a. A score of X = 56, on an exam with 50 and 4, or a score of X = 60 on an exam with 50 and 20.b A score of X = 40, on an exam with 45 and 2, or a score of X = 60 on an
21. A distribution of exam scores has a mean of 80.a. If your score is X = 86, which standard deviation would give you a better grade: 4 8?b. If your score is X = 74, which standard deviation would give you a better grade: 4 or 8?
20. For each of the following populations, would a score of X 50 be considered a central score (near the middle of the distribution) or an extreme score (far out in the tail of the distribution)?a. 45 and 10b. 45 and 2c. 90 and 20d. 60 and 20
19. In a distribution of scores, X 64 corresponds to z 1.00, and X 67 corresponds to z 2.00. Find the mean and standard deviation for the distribution.
18. In a population of exam scores, a score of X 48 corresponds to z 1.00 and a score of X 36 corresponds to z 0.50. Find the mean and standard deviation for the population. (Hint: Sketch the distribution and locate the two scores on your sketch.)
17. For a population with a mean of 70, a score of X 62 corresponds to z 2.00. What is the population standard deviation?
16. For a sample with a mean of M 45, a score of X 59 corresponds to z 2.00. What is the sample standard deviation?
15. For a sample with a standard deviation of s 10, a score of X 65 corresponds to z 1.50. What is the sample mean?
14. For a population with a standard deviation of 8, a score of X 44 corresponds to z 0.50. What is the population mean?
13. A score that is 12 points above the mean corresponds to a z-score of z 3.00. What is the population standard deviation?
12. A score that is 6 points below the mean corresponds to a z-score of z 0.50. What is the population standard deviation?
11. Find the X value corresponding to z 0.25 for each of the following distributions.a. 40 and 4b. 40 and 8c. 40 and 12d. 40 and 20
9. A sample has a mean of M 80 and a standard deviation of s 10. For this sample, find the X value corresponding to each of the following z-scores.z 0.80 z 1.20 z 2.00 z 0.40 z 0.60 z 1.80
8. A sample has a mean of M 40 and a standard deviation of s 6. Find the z-score for each of the following X values from this sample.X 44 X 42 X 46 X 28 X 50 X 37
7. A population has a mean of 40 and a standard deviation of 8.a. For this population, find the z-score for each of the following X values.X 44 X 50 X 52 X 34 X 28 X 64b. For the same population, find the score (X value)that corresponds to each of the following z-scores.z 0.75 z 1.50 z 2.00 z
6. For a population with a mean of 100 and a standard deviation of 12,a. Find the z-score for each of the following X values.X 106 X 115 X 130 X 91 X 88 X 64b. Find the score (X value) that corresponds to each of the following z-scores.z 1.00 z 0.50 z 2.00 z 0.75 z 1.50 z 1.25
5. For a population with 40 and 7, find the z-score for each of the following X values. (Note: You probably will need to use a formula and a calculator to find these values.)X 45 X 51 X 41 X 30 X 25 X 38
4. For a population with 50 and 8,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 54 X 62 X 52 X 42 X 48 X 34b. Find the score (X value)
3. A distribution has a standard deviation of 6.Describe the location of each of the following z-scores in terms of position relative to the mean. For example, z 1.00 is a location that is 6 points above the mean.a. z 2.00b. z 0.50c. z 2.00d. z 0.50
2. A distribution has a standard deviation of 12.Find the z-score for each of the following locations in the distribution.a. Above the mean by 3 points.b. Above the mean by 12 points.c. Below the mean by 24 points.d. Below the mean by 18 points.
1. What information is provided by the sign (/) of a z-score? What information is provided by the numerical value of the z-score?
23. Wegesin and Stern (2004) found greater consistency(less variability) in the memory performance scores for younger women than for older women. The following data represent memory scores obtained for two women, one older and one younger, over a series of memory trials.a. Calculate the variance of
22. In an extensive study involving thousands of British children, Arden and Plomin (2006) found significantly higher variance in the intelligence scores for males?
21. For the following sample of n 7 scores:8, 6, 5, 2, 6, 3, 5a. Sketch a histogram showing the sample distribution.b. Locate the value of the sample mean in your sketch, and make an estimate of the standard deviation (as done in Example 4.5).c. Compute SS, variance, and standard deviation for the
20. For the following population of N 6 scores:5, 0, 9, 3, 8, 5a. 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. Calculate SS, variance, and standard deviation for the following sample of n 5 scores: 9, 6, 2, 2, 6.(Note: The definitional formula works well with these scores.)
18. Calculate SS, variance, and standard deviation for the following population of N 7 scores: 8, 1, 4, 3, 5, 3, 4. (Note: The definitional formula works well with these scores.)
17. Calculate SS, variance, and standard deviation for the following population of N 8 scores: 0, 0, 5, 0, 3, 0, 0, 4. (Note: The computational formula works well with these scores.)
16. Calculate SS, variance, and standard deviation for the following sample of n 4 scores: 3, 1, 1, 1. (Note: The computational formula works well with these scores.)
15. For the data in the following sample:8, 1, 5, 1, 5a. Find the mean and the standard deviation.b. Now change the score of X 8 to X 18, and find the new mean and standard deviation.c. Describe how one extreme score influences the mean and standard deviation.
14. The range is completely determined by the two extreme scores in a distribution. The standard deviation, on the other hand, uses every score.a. Compute the range (choose either definition) and the standard deviation for the following sample of n 5 scores. Note that there are three scores
13. Calculate the mean for each of the following samples and then decide (yes or no) whether it would be easy to use the definitional formula to calculate the value for SS.Sample A: 1, 4, 7, 5 Sample B: 3, 0, 9, 4
12. There are two different formulas or methods that can be used to calculate SS.a. Under what circumstances is the definitional formula easy to use?b. Under what circumstances is the computational formula preferred?
11. For the following population of N 6 scores:11, 0, 2, 9, 9, 5a. Calculate the range and the standard deviation.(Use either definition for the range.)b. Add 2 points to each score and compute the range and standard deviation again. Describe how adding a constant to each score influences measures
10. A student was asked to compute the mean and standard deviation for the following sample of n 5 scores: 81, 87, 89, 86, and 87. To simplify the arithmetic, the student first subtracted 80 points from each score to obtain a new sample consisting of 1, 7, 9, 6, and 7. The mean and standard
9. A population has a mean of 30 and a standard deviation of 5.a. If 5 points were added to every score in the population, what would be the new values for the mean and standard deviation?b. If every score in the population were multiplied by 3 what would be the new values for the mean and
8. On an exam with a mean of M 82, you obtain a score of X 86.a. Would you prefer a standard deviation of s 2 or s 10? (Hint: Sketch each distribution and find the location of your score.)b. If your score were X = 78, would you prefer s 2 or s 10? Explain your answer.
7. A sample has a mean of M 50 and a standard deviation of s 12.a. Would a score of X 56 be considered an extreme value (out in the tail) in this sample?b. If the standard deviation were s 3, would a score of X 56 be considered an extreme value?
6. Explain what it means to say that the sample variance provides an unbiased estimate of the population variance.
5. What does it mean for a sample to have a standard deviation of s 5? Describe the scores in such a sample. (Describe where the scores are located relative to the sample mean.)
4. What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample.
3. Can SS ever have a value less than zero? Explain your answer.
2. A population has 100 and 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. SSb. Variancec. Standard deviation
26. Does it ever seem to you that the weather is nice during the work week, but lousy on the weekend?Cerveny and Balling (1998) 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. A nutritionist studying weight gain for college freshmen obtains a sample of n = 20 first-year students at the state college. Each student is weighed on the first day of school and again on the last day of the semester. The following scores measure the change in weight, in pounds, for each
24. One question on a student survey asks: In a typical week, how many times do you eat at a fast food restaurant? The following frequency distribution table summarizes the results for a sample of n 20 students.Number of times per week f 5 or more 2 4 2 3 3 2 6 1 4 0 3a. Find the mode for this
23. 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 news reporter interviewed people shopping in a local mall and asked how much they spent on summer vacations. Most people
22. Identify the circumstances in which the median rather than the mean is the preferred measure of central tendency.
21. Explain why the mean is often not a good measure of central tendency for a skewed distribution.
20. One sample has a mean of M 6 and a second sample has a mean of M 12. 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 5 scores?b. What is the mean for the combined set if the first sample has n 4 scores and
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 7 scores?b. What is the mean for the combined set if the first sample has n 3 and the
18. A population of N 16 scores has a mean of 20.After one score is removed from the population, the new mean is found to be 19. What is the value of the score that was removed? (Hint: Compare the values for X before and after the score was removed.)
17. A sample of n 7 scores has a mean of M 5. After one new score is added to the sample, the new mean is found to be M 6. What is the value of the new score? (Hint: Compare the values for X before and after the score was added.)
16. A sample of n 7 scores has a mean of M 9. One score in the sample is changed from X 19 to X 5.What is the value for the new sample mean?
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