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# Applied Math 40S March 18, 2008

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Understanding and working with (standardized) z-scores.

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### Applied Math 40S March 18, 2008

1. 1. Working with z-scores z by sandcastlematt
2. 2. HOMEWORK Four hundred people were surveyed No. of Videos Returned No. of Persons to find how many videos they had 1 28 rented during the last month. 2 102 Determine the mean and median of the 3 160 frequency distribution shown below, 4 70 and draw a probability distribution 5 25 histogram. Also, determine the mode 6 13 by inspecting the frequency 7 0 distribution and the histogram. 8 2
3. 3. HOMEWORK The table shows the weights weight interval mean interval # of infants (in pounds) of 125 newborn 3.5 to 4.5 4 4 infants. The first column 4.5 to 5.5 5 11 shows the weight interval, 5.5 to 6.5 6 19 the second column the 6.5 to 7.5 7 33 average weight within each 7.5 to 8.5 8 29 weight interval, and the third 8.5 to 9.5 9 17 column the number of 9.5 to 10.5 10 8 newborn infants at each 10.5 to 11.5 11 4 weight. Total 125 (a) Calculate the mean weight and standard deviation. (b) Calculate the weight of an infant at one standard deviation below the mean weight, and one standard deviation above the mean. (c) Determine the number of infants whose weights are within one standard deviation of the mean weight. (d) What percent of the infants have weights that are within one standard deviation of the mean weight?
4. 4. HOMEWORK The table shows the lengths in millimetres of 52 arrowheads. 16 16 17 17 18 18 18 18 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 30 30 30 30 31 33 33 34 35 39 40 (a) Calculate the mean length and the standard deviation. (b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean. (c) How many arrowheads are within one standard deviation of the mean? (d) What percent of the arrowheads are within one standard deviation of the mean length?
5. 5. Standard Score AKA z-score: A z-score indicates how many standard deviations a specific score is from the mean of a distribution. For example, the following table shows some z-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 A z-score may also be described by the following formula: z = the z-score (standardized score) x = a number in the distribution μ = the population mean σ = standard deviation for the population
6. 6. 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?
7. 7. The Canadian Armed Forces used to have a height requirement of 158 cm to 194 cm for men. The mean height of Canadian men at that time was 176 cm with a standard deviation of 8 cm. What was the z-score range for allowable heights?
8. 8. Standardizing Two Sets of Scores Two consumer groups, Vancouver Halifax one in Vancouver and Cereal Brand Rating Cereal Brand Rating one in Halifax, recently A 1 P 25 tested five brands of B 10 Q 35 breakfast cereal for taste C 15 R 45 appeal. Each consumer D 21 S 50 group used a different E 28 T 70 rating system. Use z-scores to determine which cereal has the higher taste appeal rating.
9. 9. Using Z-Score, Mean, and Standard Deviation to Calculate the Real Score 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?
10. 10. HOMEWORK A survey was conducted at DMCI to determine the number of music CDs each student owned. The results of the survey showed that the average number of CDs per student was 73 with a standard deviation of 24. After the scores were standardized, the people doing the survey discovered that DJ Chunky had a z-score rating of 2.9. How many CDs does Chunky have?
11. 11. HOMEWORK The contents in the cans of several cases of soft drinks were tested. The mean contents per can is 356 mL, and the standard deviation is 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?
12. 12. HOMEWORK 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?