z-scores z-scores z-scores and the Normal Distribu4on PSYC 300A - Lecture 3 Dr. J. Nicol

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1 z-scores and the Normal Distribu4on PSYC 300A - Lecture 3 Dr. J. Nicol z-scores Knowing a raw score does not inform us about the rela4ve loca4on of that score in the distribu4on The rela4ve loca4on of a score in a distribu4on depends on the mean and standard devia4on of the distribu4on z-scores reveal the exact loca4on of a score in a distribu4on and permit us to determine its posi4on rela4ve to the mean of the distribu4on Unlike raw scores, meaningful comparisons can be made between z-scores from different distribu4ons z-scores The popula4on mean (μ) and standard devia4on (σ) are used to transform a raw score (X) into a standardized score called a z-score z-scores are in standard devia4on units: they express scores in terms of how many standard devia4ons they are away from the mean of the distribu4on

2 z-scores A posi4ve z-score indicates that the raw score is located above the mean of the distribu4on, and a nega4ve z-score indicates that the raw score is located below the mean of the distribu4on Numerical value of a z-score indicates how many standard devia4ons (σ) the raw score (i.e., X value) lies from the mean (μ) of the distribu4on The z-score distribu4on has a μ = 0 and σ = 1 Score (X) Mean (μ) (X i - μ) (X i - μ) = 18.0 SS = 24.0 μ = 18/6 σ 2 = SS/N = 24/6 =4 σ = 4 = 2 Score (X) z-score 5 (5-3)/2 = 1 2 (2-3)/2 = (3-3)/2 = (2-3)/2 = (0-3)/2 = (6-3)/2 = 1.50 μ = z/n = 0 σ 2 = SS/N = 6/6 = 1 σ = 1 = 1

3 Standardizing a distribu4on of raw scores into z-scores does not change the shape of the distribu4on Scores remain in the same place on the z-axis as they were on the x-axis Find the z-score for a given X-value A distribu4on of scores has μ = 86 and σ = 7. What z-score corresponds to X = 95 in that distribu4on? z = (X - μ) / σ z = (95-86) / 7 z = 9/7 z = 1.29 Find the X-value for a given z-score In a normal distribu4on of scores with μ = 75 and σ = 12 what X value corresponds to z = -1.50? X = μ + (z)σ X = 75 + (-1.50)12 X = X = 57

4 Comparing Distribu4ons Comparing apples and oranges A student got 73% on her psychology exam and 78% on her biology exam In which course did she do becer rela4ve to her classmates? To compare scores from different distribu4ons first we need to standardize the scores Psychology distribu4on: X = 73%; μ = 65%; σ = 12% Biology distribu4on: X = 78%; μ = 66%; σ = 18% z-score (psychology) = (73-65)/12 = z-score (biology) = (78-66)/18 = Her grade is standard devia4ons above the mean in both courses So rela4ve to her classmates she performed equally on the two midterms Standardizing to Chosen Parameters A standardized distribu>on can be adjusted to fit predetermined parameters To give a standardized distribu4on a new μ and σ: 1) Transform the X scores into z-scores 2) Convert the z-scores back into X scores based on the new μ and σ New scores are at the same z-score loca4on in the new distribu4on as they were in original distribu4on (i.e., rela4ve loca4ons of scores do not change)

5 Distribu4on of grades has a μ = 58% and σ = 23% Make a new distribu4on with μ = 70% and σ = 15% In the new distribu4on, what are the new grades for students that had grades of 53% and 85% in the original distribu4on? z-score (53%) = (53-58)/23 = New grade = μ + z(σ) = 70 + (- 0.22)(15) = 66.7% z-score (85%) = (85-58)/23 = 1.17 New grade = μ + z(σ) = 70 + (1.17)(15) = 87.6% Probability A method for determining the likelihood of obtaining a specific sample from a specific popula4on Can be defined as a propor4on of the area under the curve of a distribu4on of scores Probability and the Normal Distribu4on The normal distribu4on can be described by the propor4on of the area under the curve that is contained in each sec4on

6 Due to the shape of the normal distribu4on, scores toward the middle have a high probability of being randomly selected and scores in the tails have a low probability of being randomly selected Density The propor4on of the area under the normal curve in sec4on A is less than the propor4on in sec4on B Density A B The probability of selec4ng a score from sec4on A is less than the probability of selec4ng a score from sec4on B μ Aier a distribu4on of scores has been standardized the normal probability distribu4on can be used to determine the probability of observing a specific score

7 Aier a distribu4on of scores has been standardized the normal probability distribu4on can be used to determine the probability of observing a specific score Density GRE scores are distributed normally with a popula4on mean of μ = 500 and standard devia4on of σ = 100 Given the known propor4ons for the normal distribu4on, what is the probability of randomly selec4ng an individual from this popula4on who has a GRE score greater than 700? z = ( ) /100 z = 2.00 In the normal distribu4on, 2.2% of the scores lie in the upper tail beyond the loca4on z = 2.00 The Unit Normal table provides a list of the propor4ons of the normal distribu4on for a full range of possible z-score values

8 The Unit Normal Table Because the normal distribu4on is symmetrical, the propor4ons are the same for the posi4ve and nega4ve values of a specific z-score Propor4ons are posi4ve and between 0.0 and 1.0 The larger por4on of the area under the normal curve is called the body and the smaller por4on of the area under the normal curve is called the tail A posi>ve z-score means that the tail of the distribu4on is on the right side and the body is on the lei A nega>ve z-score means that the tail of the distribu4on is on the lei side and the body is on the right What propor4on of the normal distribu4on corresponds to the z-score values in the shaded areas below? p = p = p = 0.309

9 What por4on of the normal distribu4on corresponds to the z-scores that are: Greater than z = 1.00? Less than z = ? Less than z = 2.25? Greater than z = -1.38? p = 0.16 p = 0.16 p = 0.99 p = 0.91 Midterm grades were normally distributed with a mean of μ = 72% and a standard devia4on of σ = 17% What is the probability of randomly selec4ng a student from the class with a grade of 75% or higher? z = (75-72) /17 z = 3/17 = 0.18 p = 0.43 The probability of randomly selec4ng a student who received a grade of at least 75% is p = 0.43 (43%) Height is a normally distributed variable In Canada, the male popula4on has a mean height of μ = 174 cm with a standard devia4on of σ = 12 cm, and the female popula4on has a mean height of μ = 163 cm with a standard devia4on of σ = 9 cm What is the probability of randomly selec4ng a male from the popula4on that is taller than 190 cm? What is the probability of randomly selec4ng a female from the popula4on that is taller than 155 cm?

10 What is the probability of randomly selec4ng a male that is 190 cm or taller? z = ( )/12 = 1.33 p = 0.09 What is the probability of randomly selec4ng a female that is 155 cm or taller? z = ( )/9 = p = 0.82 What z-score divides the normal distribu4on into: Two equal halves? z = 0.00 The upper 65% and the lower 35%? z = The upper 19% and the lower 81%? z = 0.88 The Unit Normal table also allows you to begin with a known propor4on and then look up the corresponding z-score z = 1.28 z = ± 0.84

11 What z-scores form the boundaries of the: Middle 40% of the normal distribu4on? Middle 72% of the normal distribu4on? Middle 95% of the normal distribu4on? z = ± 0.53 z = ± 1.08 z = ± 1.96 IQ scores are normally distributed with a popula4on mean of μ = 100 and a standard devia4on of σ = 15 What propor4on of the area under the curve of the distribu4on lies between IQ scores of 90 and 115? For X = 90 z = (90-100)/15 = p = 0.25 For X = 115 z = ( )/15 = 1.00 p = 0.34 Area under the curve between scores 90 and 115 p = = 0.59 IQ scores are normally distributed with a popula4on mean of μ = 100 and a standard devia4on of σ = 15 What propor4on of the area under the curve of the distribu4on lies between IQ scores of 105 and 125? For X = 105 z = ( )/15 = 0.33 p = 0.37 For X = 120 z = ( )/15 = 1.67 p = 0.05 Area under the curve between scores of X = 105 and X = 115 p = = 0.32

12 The distribu4on of commu4ng 4mes for Canadian workers is normal with mean of μ = 24.3 minutes and a standard devia4on of σ = 10 minutes What range of commute 4mes defines the middle 90% of the distribu4on? 90% / 2 = 45% z = ± (σ) = 1.65(10) = 16.5 X (upper) = = 40.8 X (lower) = = % of Canadian commuters spend between 7.8 and 40.8 minutes commu4ng to work each day Percen4les A score referred to by its rank is called a percen>le The percen4le rank in a normal distribu4on is the propor4on to the lei of the score (i.e., the percentage/ propor4on of scores at or below that value)

13 MCAT scores are normally distributed with a popula4on mean of μ = 300 and a standard devia4on of σ = 60 Medical schools are only accep4ng applica4ons from students that scored in at least the 80 th percen4le What MCAT score would you need to get in order to be eligible to apply to medical school? 80 th percen4le: z = = (X - 300)/60 X = X = 351 The 80 th percen4le of the MCAT corresponds to a score of 351, so test takers with MCAT scores lower than that are not eligible to apply to medical school Probability is used to predict the type of samples that are likely to be obtained from a specific popula4on Inferen4al sta4s4cs rely on the connec4on between samples and popula4ons when using sample data to make conclusions about popula4ons

14 A sample is selected from the popula4on and receives a treatment. Research ques4on is: does the treatment have an effect? Probability can be used to evaluate a treatment effect by iden4fying likely and very unlikely outcomes Boundaries at z = ± 1.96 provide objec4ve criteria for deciding whether a sample provides evidence of a treatment effect When an obtained sample is located in the tail beyond one of the z = ± 1.96 boundaries, then we can conclude: It is an extreme value (i.e., nearly 2 standard devia4ons from the mean), and so is very different from most samples in the original (untreated) popula4on It is a very unlikely value with a very low probability of being obtained if the treatment has no effect Therefore, the sample provides strong evidence that the treatment has had a significant effect