PSYC212 Handout 4 z score packet (sh vrs) PDF

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z-score Dr. Syed

z-score packet

Formula to convert a score (X) to a z score when M, SD of the sample are known.

𝑿−𝑴 𝒛= 𝑺𝑫

Formula to convert a z score to a score (X) when M, SD of the sample are known.

𝑿 = (𝒛)(𝑺𝑫) + 𝑴 Always plot the z score or scores on a normal distribution. Be sure to pay attention to the sign of the z-score. A positive z score indicates that it is above the mean. A negative z score indicates that it is below the mean. Using the z table, interpret the scores. Note that the z table has no positive or negatives because they are the mirror image of one another. This is why it is so important to plot your scores on a curve before you use the z table so that you know what direction the proportions from the table should be positioned. The columns of the z table: A: this is the actual z score without the sign B: this is the proportion of scores between the score and the mean (i.e. z = 0), for ease of interpretation you can convert the proportions (decimal format) to percentages by multiplying them by 100. C: this is the proportion of scores between the score and the closest tail. When z is negative “c” is in the left tail of the distribution When z is positive “c” is in the right tail of the distribution 1|Page

z-score Dr. Syed

Empirical Rule The empirical rule is the statistical rule stating that for a normal distribution, almost all data will fall within three standard deviations of the mean.

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1. A psychologist is interested in evaluating a child’s IQ and Achievement Scores. IQ scores are a measure of innate ability whereas achievement scores represent what a child has learned over time. o The child scored 125 on the IQ Test o The same child scored 1150 on the Achievement Test o The mean and standard deviation for the IQ test are as follows: (M = 110, SD = 15) o The mean and standard deviation for the Achievement test are as follows: (M = 1200, SD = 200) a. b. c. d.

Compute the z-scores Plot and Interpret the scores. What is the percentile rank of each score? What percentage of scores are between these two scores?

2. Image you were offered two jobs. Based on the information below, which job would you choose: o City A o Cost of Living: : (M = $50, 000, SD = $15,000) o Salary offered: $27,000 o City B o Cost of Living: : (M = $14,000, SD = $1,000) o Salary offered: $12,000 a. b. c. d.

Compute the z-scores Plot and Interpret the scores. What is the percentile rank of each score? What percentage of scores are between these two scores?

3. If the mean SAT score is 1000 with SD = 200, what score do I have to get in order to be in the top 10% of people who took the test?

4. Women’s heights have a mean of 63.6 in. and a standard deviation of 2.5 inches. Find the z score corresponding to a woman with a height of 70 inches. Next, determine whether the height is unusual.

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z-score Dr. Syed

5. Mike got an 88 on his math test, and Joy got an 86. Who did better? In Mike’s class, M = 87 and SD = 4. In Joy’s class, M = 83 and SD = 3. a. Compute the z-scores b. Plot and Interpret the scores. c. What is the percentile rank of each score? d. What percentage of scores are between these two scores? e. Who had the higher raw score? Who had the higher standardized score? Explain the difference between your responses. 6. Three students took a Writing Test (M = 15, SD = 5) and Reading Test (M = 100, SD = 10). o Tom (Writing Score 12, Reading Score 100) o Susan (Writing Score 13, Reading Score 92) o Martin (Writing Score 10, Reading Score 115) o Find the z-scores for each student’s writing and reading scores. o Find the percentile for each student’s writing and reading scores. o For each student determine if they have a relatively better writing or reading score. 7. A psychologist gave two of his patients a depression scale. The psychologist is interested in the percentage of patients who fall between their two scores. o The first patient scored 125 o The second patient scored 98 o The mean and standard deviation for the Depression scale is as follows: (M = 100, SD = 20) o Find the percentage of scores between 98 – 125. o What is the percentile rank of each patient’s score? The same two patients took an anxiety test. o The first patient scored 1200 o The second patient scored 900 o The mean and standard deviation for the anxiety test is as follows: (M = 1050, SD = 250) o Find the percentage of scores between 900 - 1200. o What is the percentile rank of each patient’s score? o Evaluate the relative standing of each patient’s depression and anxiety. Interpret the results.

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8. A fifth grader takes a standardized achievement test (M = 125, SD = 15) and scores a 148. What is the child’s percentile? 9. Three students take equivalent stress tests. Which is the highest relative score? a. A score of 144 on a test with a mean of 128 and a standard deviation of 34. b. A score of 90 on a test with a mean of 86 and a standard deviation of 18. c. A score of 18 on a test with a mean of 15 and a standard deviation of 5. 10. A normal distribution of scores has a standard deviation of 10. Find the z-scores corresponding to each of the following values: a) b) c) d)

A score that is 20 points above the mean. A score that is 10 points below the mean. A score that is 15 points above the mean A score that is 30 points below the mean.

11. For each z-score below, find the percentile (percent of individuals scoring at or below): a) z = – 0.47 b) z = 2.24 12. For each z-score below, find the proportion of cases falling above the z: a) z = 0.24 b) z = – 2.07 13. For each z-score below, find the percentage between the mean and the z-score: a) z = 1.17 b) z = – 1.37 14. A patient recently diagnosed with Alzheimer’s disease takes a cognitive abilities test and scores a 45. The mean on this test is 52 and the standard deviation is 5. What is the patient’s percentile? 15. Pat and Chris both took a spatial abilities test (M = 80, SD = 8). Pat scores a 76 and Chris scored a 94. What percentage of individuals would score between Pat and Chris? 16. Scores on the SAT form a normal distribution with and M = 500, SD = 100 a) What is the minimum score necessary to be in the top 15% of the SAT distribution?

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17. The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of 35 and a standard deviation of 6. Assuming these raw scores form a normal distribution: a) What proportion of raw scores are between 28 and 38? b) What proportion of raw scores are between 41 and 44? c) What score would one need to be in the top 10%? 18. A set of reading comprehension scores for fourth-grade students has a mean of 25 and a standard deviation of 5. A set of reading comprehension scores for ninth-grade students has a mean of 30 and a standard deviation of 10. Cameron (a fourth grader) scored 30 on his reading comprehension test. His brother Ron (a ninth grader) scored 35 on his reading comprehension test. a) Convert their scores to z-scores and plot on a distribution. b) Convert their scores to percentiles. c) Who scored relatively better on the reading comprehension test? Explain. d) What is the minimum reading comprehension score a student would need to score in the top 10% of the fourth grade? e) What is the minimum reading comprehension score a student would need to score in the top 10% of the ninth grade? f) What percentage of scores were between Cameron and Ron’s scores?

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