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# Standard scores

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### Standard scores

1. 1. Applied 40S Standard Scores (z-scores) A standard score indicates how many standard deviations a specific score is from the mean of a distribution. For example, the following table shows some standard scores for a distribution with:  a mean of 150  a standard deviation of 10 actual score 130 140 150 160 170 174 180 standard score -2.0 -1.0 0.0 1.0 2.0 2.4 3.0 The term used to describe the above standardized score is z-score. The z-score may also be described by the following formula: where: z = the z-score (standardized score) x = a number in the distribution μ= the population mean σ= standard deviation for the population Example 1: Student A from Parkland High and Student B from Metro Collegiate both had a final math mark of 95 percent. The awards committee must select the student with the 'highest' mark for the annual math award. The table shows the mean mark and standard deviation for each school. School Mean Standard Deviation Parkland High 75 8 Metro Collegiate 77 6 Calculate the z-score for each student. Which student should receive the award?
2. 2. Student A: Student B: Student A's 95 percent mark was 2.5 standard deviations above the class mean at Parkland High, and Student B's 95 percent mark was 3.0 standard deviations above the class mean at Metro Collegiate. Therefore, Student B has a higher z-score, and should receive the award. Example 2: Numerous packages of raisins were weighed. The mean mass was 1600 grams, and the standard deviation was 40 grams. Trudy bought a package that had a z-score of -1.6. What was the mass of Trudy's package of raisins? For the formula, z = -1.6 m = 1600 s = 40 x = ? Use algebra to solve for 'x': x = 1600 + (40)(-1.6) x = 1536 Therefore, the mass of Trudy's package is 1536 grams.
3. 3. Assignment 1. The contents in the cans of several cases of soft drinks were tested. The mean contents per can was 356 mL, and the standard deviation was 1.5 mL. a. Two cans were randomly selected and tested. One can held 358 mL, and the other can 352 mL. Calculate the z-score of each. b. Two other cans had z-scores of -3 and 1.85. How many mL did each contain? 2. North American women have a mean height of 161.5 cm and a standard deviation of 6.3 cm. a. A car designer designs car seats to fit women taller than 159.0 cm. What is the z-score of a woman who is 159.0 cm tall? b. The manufacturer designs the seats to fit women with a maximum z-score of 2.8. How tall is a woman with a z-score of 2.8? 3. The following information concerning grades is posted on the bulletin board. Test Grade Z-Score 55 -2 65 -1 75 0 85 1 95 2 a. What is the average test grade? b. What test score has a z-score of -2.76? 4. Tammy and Jamey both applied for the same job. Tammy scored 80 on a provincial aptitude test where the mean was 70 and the standard deviation was 4.2. Jamey scored 510 on the company exam where the mean was 490 and the standard deviation was 10.3. Assuming the company uses these test results as the only criteria for hiring new employees and that both tests are considered to be equal by company officials, who might get the job? Explain your answer.
4. 4. 5. Mary Reed recently entered college and has not decided on a major. She decides to take an aptitude test in order to help her select a major. Her results, as well as the results of the other candidates, are as follows: Talent Mean Standard Deviation Mary Reed's Score Writing 58 3.7 46 Acting 84 7.9 85 Medicine 37 2.8 44 Law 49 4.7 41 a. Rewrite each of Mary Reed's scores as z-scores. b. In which area does she have the most talent? c. In which area does she have the least talent? Answers: 1. a) 1st can 1.33, 2nd can -2.67 b)1st can 351.5mL, 2nd can 358.8 mL 2. a) -0.40 b)179.1 cm 3. a) 75 b)47.4 4. Tammy 2.38, Jamey 1.94, Tammy should get the job. 5. a) writing -3.24, acting 0.13, medicine 2.50, law -1.70 b) medicine c) writing