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essentials of statistics
Questions and Answers of
Essentials Of Statistics
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
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
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
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
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
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
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
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%
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
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
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
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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.
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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?
15. A population of N 20 scores has a mean of 15.One score in the population is changed from X 8 to X 28. What is the value for the new population mean?
14. A sample of n = 9 scores has a mean of M 10. One person with a score of X 2 is removed from the sample. What is the value for the new sample mean?
13. A sample of n 11 scores has a mean of M 4. One person with a score of X 16 is added to the sample.What is the value for the new sample mean?
12. A sample of n 5 scores has a mean of M 12. If one person with a score of X 8 is removed from the sample, what will be the value for the new mean?
11. A sample of n 8 scores has a mean of M 10. If one new person with a score of X 1 is added to the sample, what will be the value for the new mean?
10. A population with a mean of 10 has X 250.How many scores are in the population?
9. A sample of n 7 scores has a mean of M 9. What is the value of X for this sample?
8. In 2007, professional golfer Tiger Woods competed in 16 PGA golf tournaments. He finished 1st seven times, 2nd three times, and one time each in 6th, 9th, 12th, 15th, 22nd, and 37th. Find his
7. For the following samplea. Assume that the scores are measurements of a continuous variable and find the median by locating the precise midpoint of the distribution.b. Assume that the scores are
6. Find the mean, median, and mode for the scores in the following frequency distribution table:X f 10 1 9 2 8 3 7 3 6 4 5 2
5. Find the mean, median, and mode for the scores in the following frequency distribution table:X f 8 1 7 4 6 2 5 2 4 2 3 1
4. Find the mean, median, and mode for the following sample of scores:8, 7, 8, 8, 4, 9, 10, 7, 8, 8, 9, 8
3. Find the mean, median, and mode for the following sample of scores:6, 2, 4, 1, 2, 2, 3, 4, 3, 2
2. Why is it necessary to have more than one method for measuring central tendency?
1. What general purpose is served by a good measure of central tendency?
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