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1. In words, explain what is measured by each of the following: a. SS b. Variance c. Standard deviation 2. Can SS ever have a value less than zero? Explain your answer. 3. Is it possible to obtain a negative value for the variance or the standard deviation? 4. What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample. 5. Explain why the formulas for sample variance and population variance are different. 6. A population has a mean of _ _ 80 and a standard deviation of _ _ 20. a. Would a score of X _ 70 be considered an extreme value (out in the tail) in this sample? b. If the standard deviation were _ _ 5, would a score of X _ 70 be considered an extreme value? 7. On an exam with a mean of M _ 78, you obtain a score of X _ 84. 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 _ 72, would you prefer s _ 2 or s _ 10? Explain your answer. 8. 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 standard deviation? 9. a. After 3 points have been added to every score in a sample, the mean is found to be M _ 83 and the standard deviation is s _ 8. What were the values for the mean and standard deviation for the original sample? b. After every score in a sample has been multiplied by 4, the mean is found to be M _ 48 and the standard deviation is s _ 12. What were the values for the mean and standard deviation for the original sample? 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 deviation for the new sample were then calculated to be M _ 6 and s _ 3. What are the values of the mean and standard deviation for the original sample? 11. For the following population of N _ 6 scores: 11, 0, 2, 9, 9, 5 a. 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 of variability. 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? 13. Calculate the mean and SS (sum of squared deviations) for each of the following samples. Based on the value for the mean, you should be able to decide which SS formula is better to use. Sample A: 1, 4, 8, 5 Sample B: 3, 0, 9, 4 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 clustered around the mean in the center of the distribution, and two extreme values. Scores: 0, 6, 7, 8, 14. b. Now we break up the cluster in the center of the distribution by moving two of the central scores out to the extremes. Once again compute the range and the standard deviation. New scores: 0, 0, 7, 14, 14. c. According to the range, how do the two distributions compare in variability? How do they compare according to the standard deviation? 15. For the data in the following sample: 8, 1, 5, 1, 5 a. 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. 16. Calculate SS, variance, and standard deviation for the following sample of n _ 4 scores: 7, 4, 2, 1. (Note: The computational 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.) 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.) 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.) 20. For the following population of N _ 6 scores: 3, 1, 4, 3, 3, 4 a. 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 for the population. (How well does your estimate compare with the actual value of _?) 21. For the following sample of n _ 7 scores: 8, 6, 5, 2, 6, 3, 5 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 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 the actual value of s?) 22. In an extensive study involving thousands of British children, Arden and Plomin (2006) found significantly higher variance in the intelligence scores for males than for females. Following are hypothetical data, similar to the results obtained in the study. Note that the scores are not regular IQ scores but have been standardized so that the entire sample has a mean of M _ 10 and a standard deviation of s _ 2. a. Calculate the mean and the standard deviation for the sample of n _ 8 females and for the sample of n _ 8 males. b. Based on the means and the standard deviations, describe the differences in intelligence scores for males and females. Female Male 98 11 10 10 11 13 12 86 9 10 11 14 99 23. In the Preview section at the beginning of this chapter we reported a study by Wegesin and Stern (2004) that 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 the scores for each woman. b. Are the scores for the younger woman more consistent (less variable)? Younger Older 87 65 68 75 87 76 88 85 1. What information is provided by the sign (_/–) of a z-score? What information is provided by the numerical value of the z-score? 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. 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__2.00 b. z__0.50 c. z _ –2.00 d. z _ –0.50 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 _ 34 b. Find the score (X value) that corresponds to each of the following z-scores. (Again, you should be able to find these values without any formula or serious calculations.) z _ 1.00 z _ 0.75 z _ 1.50 z _ –0.50 z _ –0.25 z _ –1.50 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 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 _ 64 b. 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 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 _ 64 b. 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 _ –0.25 z _ –0.50 z _ 1.25 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 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 10. Find the z-score corresponding to a score of X _ 60 for each of the following distributions. a. _ _ 50 and _ _ 20 b. _ _ 50 and _ _ 10 c. _ _ 50 and _ _ 5 d. _ _ 50 and _ _ 2 11. Find the X value corresponding to z _ 0.25 for each of the following distributions. a. _ _ 40 and _ _ 4 b. _ _ 40 and _ _ 8 c. _ _ 40 and _ _ 12 d. _ _ 40 and _ _ 20 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? 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? 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? 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? 16. For a sample with a mean of _ _ 45, a score of X _ 59 corresponds to z _ 2.00. What is the sample standard deviation? 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? 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.) 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. 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 _ _ 10 b. _ _ 45 and _ _ 2 c. _ _ 90 and _ _ 20 d. _ _ 60 and _ _ 20 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? 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 exam with _ _ 70 and _ _ 20. c. A score of X _ 62, on an exam with _ _ 50 and _ _ 8; or a score of X _ 23 on an exam with _ _ 20 and _ _ 2. 23. A distribution with a mean of _ _ 62 and a standard deviation of _ _ 8 is 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 _ 60 b. X _ 54 c. X _ 72 d. X _ 66 24. A distribution with a mean of _ _ 56 and a standard deviation of _ _ 20 is 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 _ 46 b. X _ 76 c. X _ 40 d. X _ 80 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 _ _ 100 and a standard deviation of _ _ 20. 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. 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 20%, 5 for 30%, 3 for 40%, 1 for 50%, and 1 for 100% off. Assuming that all 50 sections are equally likely, a. What is the probability that a customer’s purchase will be free (100% off)? b. What is the probability that a customer will get no savings from the wheel (0% off)? c. What is the probability that a customer will get at least 20% off? 2. A psychology class consists of 14 males and 36 females. If the professor selects names from the class list using random sampling, a. What is the probability that the first student selected will be a female? b. If a random sample of n _ 3 students is selected and the first two are both females, what is the probability that the third student selected will be a male? 3. What are the two requirements that must be satisfied for a random sample? 4. What is sampling with replacement, and why is it used? 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 _ 2.00 b. z _ 0.60 c. z _ –1.30 d. z _ –0.30 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 _ 2.20 b. z _ 1.60 c. z _ –1.50 d. z _ –0.70 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) 8. What proportion of a normal distribution is located between each of the following z-score boundaries? a. z _ –0.50 and z__0.50 b. z _ –0.90 and z__0.90 c. z _ –1.50 and z__1.50 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) 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 left b. 40% in the tail on the right c. 75% in the body on the left d. 99% in the body on the right 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. 12. For a normal distribution with a mean of μ _ 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. 13. A normal distribution has a mean of μ _ 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 _ 53 b. X _ 44 c. X _ 68 d. X _ 38 14. IQ test scores are standardized to produce a normal distribution with a mean of μ _ 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 greater than 140 b. Very superior intelligence: IQ between 120 and 140 c. Average or normal intelligence: IQ between 90 and 109 15. The distribution of scores on the SAT is approximately normal with a mean of μ _ 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 minimum SAT score needed to be in the highest 10% of the population? d. If the state college only accepts students from the top 60% of the SAT distribution, what is the minimum SAT score needed to be accepted? 16. The distribution of SAT scores is normal with μ _ 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 10% of the distribution from the rest? c. What SAT score, X value, separates the top 2% of the distribution from the rest? 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 μ _ 60 minutes of television each day. Assuming that the distribution of television-watching times is normal with a standard deviation of _ _ 20 minutes, find each of the following proportions. a. What proportion of 2-year-old children watch more than 90 minutes of television each day? b. What proportion of 2-year-old children watch less than 20 minutes a day? 18. Information from the Department of Motor Vehicles indicates that the average age of licensed drivers is μ _ 45.7 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 older than 50 years old? b. What proportion of licensed drivers are younger than 30 years old? 19. A consumer survey indicates that the average household spends μ _ $185 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 $200 per week on groceries? b. What is the probability of randomly selecting a family that spends less than $150 per week on groceries? c. How much money do you need to spend on groceries each week to be in the top 20% of the distribution? 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 μ _ 19.8 minutes with a standard deviation of _ _ 7.2 minutes. Assuming that the distribution is approximately normal, find each of the following probabilities. a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? In symbols, p(X _ 25) _ ? b. What is the probability of randomly selecting a student with a time greater than 30 minutes? In symbols, p(X _ 30) _ ? c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what proportion of the freshmen would fail? 21. Rochester, New York, averages μ _ 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 of all purchases made in December, if Rochester finishes the month with more than 3 feet (36 inches) of total snowfall. What is the probability that the jewelry store will have to pay off on its promise? 22. A multiple-choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers, a. What is the probability of guessing correctly for any question? b. On average, how many questions would a student get correct for the entire test? c. What is the probability that a student would get more than 15 answers correct simply by guessing? d. What is the probability that a student would get 15 or more answers correct simply by guessing? 23. A true/false test has 40 questions. If a students is simply guessing at the answers, a. What is the probability of guessing correctly for any one question? b. On average, how many questions would the student get correct for the entire test? c. What is the probability that the student would get more than 25 answers correct simply by guessing? d. What is the probability that the student would get 25 or more answers correct simply by guessing? 24. A roulette wheel has alternating red and black numbered slots into one of which the ball finally stops to determine the winner. If a gambler always bets on black to win, what is the probability of winning at least 24 times in a series of 36 spins? (Note that at least 24 wins means 24 or more.) 25. One test for ESP involves using Zener cards. Each card shows one of five different symbols (square, circle, star, cross, wavy lines), and the person being tested has to predict the shape on each card before it is selected. Find each of the probabilities requested for a person who has no ESP and is just guessing. a. What is the probability of correctly predicting 20 cards in a series of 100 trials? b. What is the probability of correctly predicting more than 30 cards in a series of 100 trials? c. What is the probability of correctly predicting 50 or more cards in a series of 200 trials? 26. A trick coin has been weighted so that heads occurs with a probability of p _ _2 3_, and p(tails) _ _1 3_. If you toss this coin 72 times, a. How many heads would you expect to get on average? b. What is the probability of getting more than 50 heads? c. What is the probability of getting exactly 50 heads? 27. For a balanced coin: a. What is the probability of getting more than 30 heads in 50 tosses? b. What is the probability of getting more than 60 heads in 100 tosses? c. Parts a and b both asked for the probability of getting more than 60% heads in a series of coin tosses (_35 00 _ __1 6 0 0 0_ _ 60%). Why do you think the two probabilities are different? 28. A national health organization predicts that 20% of American adults will get the flu this season. If a sample of 100 adults is selected from the population, a. What is the probability that at least 25 of the people will be diagnosed with the flu? (Be careful: “at least 25” means “25 or more.”) b. What is the probability that fewer than 15 of the people will be diagnosed with the flu? (Be careful: “fewer than 15” means “14 or less.”