1. COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Test Review: Ch. 4-6 Peer Tutor Slides Instructor: Mr. Ethan W. Cooper, Lead Tutor © 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE EMAIL MR. COOPER AT THE FOLLWING: coopere07@students.ecu.edu

2. Chapter 4: Key Concepts Know that this chapter is on variability, which measures the differences between scores and describes the degree to which the scores are spread out or clustered together. Variability also measures how well an individual score represents the entire distribution. The 3 measures of variability: the range, standard deviation, and the variance.

3. Chapter 4: The Range The range is the distance covered by the scores in a distribution, from the smallest to the largest score. The range is the distance covered by the scores in a distribution, from the smallest to the largest score. There are 2 formulas for the range that you should be aware of for the test: There are 2 formulas for the range that you should be aware of for the test: range = URL for X max – LRL for X min range = URL for X max – LRL for X min range = X max – X min range = X max – X min This is the definition used by many computer programs. Remember: URL stands for upper real limit, and LRL stands for lower real limit.

4. Chapter 4: The Range Find the range for the following set of scores using both formulas: Find the range for the following set of scores using both formulas: 8, 1, 5, 1, 5 8, 1, 5, 1, 5

5. Chapter 4: The Range range = URL for X max – LRL for X min range = URL for X max – LRL for X min range = 8.5 – 0.5 = 8 range = 8.5 – 0.5 = 8 range = X max – X min range = X max – X min range = 8 – 1 = 7 range = 8 – 1 = 7

6. Chapter 4: Sum of Squares Use when mean is a whole number. Use for fractional means.

7. Chapter 4: Variance

8. Chapter 4: Standard Deviation

9. Find the standard deviation for the following population of N =7 scores: Find the standard deviation for the following population of N =7 scores: 8, 1, 4, 3, 5, 3, 4 8, 1, 4, 3, 5, 3, 4

10. Chapter 4: Standard Deviation X(X - µ)(X - µ) 2 88 – 4 = 0(4) 2 = 16 11 – 4 = -3(-3) 2 = 9 44 – 4 = 0(0) 2 = 0 33 – 4 = -1(-1) 2 = 1 55 – 4 = 1(1) 2 = 1 33 – 4 = -1(-1) 2 = 1 44 – 4 = 0(0) 2 = 0 SS = 8 + 9 + 0 + 1 + 1 + 1 + 0 = 28 Because the mean is a whole number.

11. Chapter 4: Standard Deviation

12. Chapter 4: Sample Standard Deviation A sample statistic is unbiased if the average value of the statistic is equal to the population parameter. A sample statistic is unbiased if the average value of the statistic is equal to the population parameter. A sample statistic is biased if the average value of the statistic either overestimates or underestimates the corresponding population parameter. A sample statistic is biased if the average value of the statistic either overestimates or underestimates the corresponding population parameter. To avoid bias when calculating s 2 or s, we use n – 1, instead of n. To avoid bias when calculating s 2 or s, we use n – 1, instead of n. n – 1 is referred to as degrees of freedom, or df. n – 1 is referred to as degrees of freedom, or df.

13. Chapter 4: Sample Standard Deviation Find the standard deviation for the following sample of n = 4 scores: Find the standard deviation for the following sample of n = 4 scores: 7, 4, 2, 1 7, 4, 2, 1

14. Chapter 4: Sample Standard Deviation XX2X2 7(7) 2 = 49 4(4) 2 = 16 2(2) 2 = 4 1(1) 2 = 1 Because the mean contains decimals.

15. Chapter 4: Sample Standard Deviation Use n-1 in the denominator because we’re working with a sample.

16. Chapter 4: More About Standard Deviation It should be noted that adding a constant to each score in a distribution does not change the standard deviation. Remember that standard deviation is a measure of variability, or the distance between scores. If the same number were added to every score, the entire distribution would shift to the right, but the distance between each score would remain the same. It should be noted that adding a constant to each score in a distribution does not change the standard deviation. Remember that standard deviation is a measure of variability, or the distance between scores. If the same number were added to every score, the entire distribution would shift to the right, but the distance between each score would remain the same.

17. Chapter 4: More About Standard Deviation However, multiplying every score by a constant would cause the standard deviation to be multiplied by the same constant. However, multiplying every score by a constant would cause the standard deviation to be multiplied by the same constant. Imagine a distribution that included scores of X = 10 and X = 11. The distance between these scores is only 1 point. Imagine a distribution that included scores of X = 10 and X = 11. The distance between these scores is only 1 point. Now imagine we multiply every score in this distribution by 2 points. Our new scores would be X = 20 and X = 22, 2 points apart. Now imagine we multiply every score in this distribution by 2 points. Our new scores would be X = 20 and X = 22, 2 points apart. Because the distance between scores changes when we multiply by a constant, the standard deviation also changes by the same constant. Because the distance between scores changes when we multiply by a constant, the standard deviation also changes by the same constant.

18. Chapter 5: Key Concepts z-Scores specify the precise location of each X value within a distribution. z-Scores specify the precise location of each X value within a distribution. The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative). The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score describes the distance from the mean by counting the number of standard deviations between X and µ. The numerical value of the z-score describes the distance from the mean by counting the number of standard deviations between X and µ. The z-score distribution will always have a mean of µ = 0 and a standard deviation of σ = 1. The z-score distribution will always have a mean of µ = 0 and a standard deviation of σ = 1.

19. Chapter 5: z-Scores

20. For a population with µ = 50 and σ = 8, find the z-score for each of the following X values: For a population with µ = 50 and σ = 8, find the z-score for each of the following X values: X = 54 X = 54 X = 42 X = 42 X = 62 X = 62 X = 48 X = 48

21. Chapter 5: z-Scores

22. Chapter 5: Find X Find the X value that corresponds with the following z- scores: (µ = 50 and σ = 8) Find the X value that corresponds with the following z- scores: (µ = 50 and σ = 8) z = -0.50 z = -0.50 z = 0.75 z = 0.75 z = -1.50 z = -1.50 z = 0.25 z = 0.25

23. Chapter 5: Find X

24. Chapter 5: Standardized Distributions Standardized distributions are composed of scores that have been transformed to create predetermined values for µ and σ. Standardized distributions are composed of scores that have been transformed to create predetermined values for µ and σ. Standardized distributions are used to make dissimilar distributions comparable. Standardized distributions are used to make dissimilar distributions comparable. A z-score distribution is an example of a standardized distribution. A z-score distribution is an example of a standardized distribution.

25. Chapter 5: Standardized Distributions A distribution with μ = 62 and σ = 8 is transformed into a standardized distribution with μ = 100 and σ = 20. Find the new, standardized score for each of the following X values: A distribution with μ = 62 and σ = 8 is transformed into a standardized distribution with μ = 100 and σ = 20. Find the new, standardized score for each of the following X values: X = 60 X = 60 X = 54 X = 54 X = 72 X = 72 X = 66 X = 66

26. Chapter 5: Standardized Distributions

29. Chapter 6: Key Concepts

30. Chapter 6: Random Sampling Sampling with replacement requires that selected individuals be returned to the population before the next selection is made. Sampling with replacement requires that selected individuals be returned to the population before the next selection is made. This ensures that the probability of selection remains constant from one selection to the next. This ensures that the probability of selection remains constant from one selection to the next. Unless otherwise specified, random sampling assumes replacement. Unless otherwise specified, random sampling assumes replacement. Probability does not remain constant when sampling without replacement. Probability does not remain constant when sampling without replacement.

31. Chapter 6: Random Sampling A psychology class consists of 14 males and 36 females. If the professor selects names from the class list using random sampling, 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?

32. Chapter 6: Random Sampling a) p = 36/50 = 0.72 b) p = 14/50 = 0.28 Because this is a random sample, replacement is assumed. Therefore, the probability of selection remains constant.

33. Chapter 6: The Normal Distribution When dealing with probability in this chapter, we are dealing with normal distributions. When dealing with probability in this chapter, we are dealing with normal distributions. The unit normal table lists proportions of the normal distribution for a full range of possible z-score values. The unit normal table lists proportions of the normal distribution for a full range of possible z-score values. Percentage of the population located between z = 0 and Z = 1. Normal distributions are symmetrical. 34.13% 13.59% 2.28%

34. Chapter 6: The Unit Normal Table The first column (A) in the table lists z-scores corresponding to different positions in a normal distribution. The first column (A) in the table lists z-scores corresponding to different positions in a normal distribution. Column B presents the proportion in the body. Column B presents the proportion in the body. The body is always the larger part of the distribution. The body is always the larger part of the distribution. Column C presents the proportion in the tail. Column C presents the proportion in the tail. The tail is always the smaller part of the distribution. The tail is always the smaller part of the distribution. Column D identifies the proportion located between the mean and the z-score. Column D identifies the proportion located between the mean and the z-score. Note: The normal distribution is symmetrical, so the proportions on the right are the same as the proportions on the left. And although the z-score values change signs (+ and -), the proportions are always positive.

35. Chapter 6: The Unit Normal Table Find each of the probabilities for a normal distribution: Find each of the probabilities for a normal distribution: p(z > 0.25) p(z > 0.25) p(z > -0.75) p(z > -0.75) p(z < 1.20) p(z < 1.20) p(-0.25 < z < 0.25) p(-0.25 < z < 0.25) p(-1.25 < z < 0.25) p(-1.25 < z < 0.25)

36. Chapter 6: The Unit Normal Table p(z > 0.25) p(z > 0.25) p(z > -0.75) p(z > -0.75) p(z < 1.20) p(z < 1.20) p(-0.25 < z < 0.25) p(-0.25 < z < 0.25) p(-1.25 < z < 0.25) p(-1.25 < z < 0.25) p = 0.4013 p = 0.7734 p = 0.8849 p = 0.0987 p = 0.4931 Find the proportion in column D for both z-scores and add them together.

37. Chapter 6: Binomial Distributions When a variable is measured on a scale consisting of exactly two categories, the resulting data are called binomial. When a variable is measured on a scale consisting of exactly two categories, the resulting data are called binomial. The normal distribution can be used to compute probabilities with binomial data. The normal distribution can be used to compute probabilities with binomial data. The two categories are defined as A and B. The two categories are defined as A and B. The probabilities associated with each category are: The probabilities associated with each category are: p = p(A) = the probability of A p = p(A) = the probability of A q = p(B) = the probability of B q = p(B) = the probability of B The number of individuals or observations in the sample is identified by n. The number of individuals or observations in the sample is identified by n. The variable X refers to the number of times category A occurs in the sample. The variable X refers to the number of times category A occurs in the sample. p + q = 1.00

38. Chapter 6: Binomial Distributions

39. Notice that each X value is represented by a bar in the histogram. This means a score of X = 8 spans the interval of 7.5 to 8.5.

40. Chapter 6: Binomial Distributions A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers, 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 than15 answers correct simply by guessing? d) What is the probability that a student would get 15 or more answers correct simply by guessing?

41. Chapter 6: Binomial Distributions a) p = ¼ = 0.25 b) pn = 0.25(48) = 12

42. Chapter 6: Binomial Distributions Both are greater than 10, so our distribution approximates a normal distribution. URL because we are excluding the score X = 15.

43. Chapter 6: Binomial Distributions LRL because we are including the score X = 15.