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# Applied 40S March 25, 2009

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Working with z-scores.

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### Applied 40S March 25, 2009

1. 1. Working with z-scores z by sandcastlematt
2. 2. Let's apply what we've learned ... mark interval mark # of students The frequency distribution table 29 to 37 33 1 at right shows the midterm marks 38 to 46 42 4 47 to 55 51 12 of 85 Grade 12 math students. 56 to 64 60 18 The ﬁrst column shows the mark 65 to 73 69 24 interval, the second column the 74 to 82 78 16 average mark within each mark 83 to 91 87 7 interval, and the third column the 92 to 100 96 3 number of students at each mark. Total 85 This is the (a) Calculate the mean and std. dev. to two decimal places. correct way Mean = 66.99 σ = 13.27 to do this. (b) Calculate the number of students that have marks within one std. dev. of the mean. 58 or 70 (c) What percent of students have marks within one std. dev. of the mean? 68% or 82%
3. 3. HOMEWORK An experiment was performed to determine the approximate mass of a penny. Three hundred pennies were weighed, and the weights were recorded in the frequency distribution table shown below. Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 Frequency 2 4 34 71 94 74 17 4 Determine the mean, median, mode, range, and standard deviation of the data. Create a histogram that shows the frequencies of different masses of this set of pennies.
4. 4. HOMEWORK No. of Videos Four hundred people were No. of Persons Returned surveyed to ﬁnd how many videos 1 28 they had rented during the last 2 102 month. Determine the mean and 3 160 4 70 median of the frequency 5 25 distribution shown below, and draw 6 13 a probability distribution 7 0 histogram. Also, determine the 8 2 mode by inspecting the frequency distribution and the histogram.
5. 5. HOMEWORK weight interval mean interval # of infants The table shows the 3.5 to 4.5 4 4 weights (in pounds) of 125 4.5 to 5.5 5 11 newborn infants. The ﬁrst 5.5 to 6.5 6 19 6.5 to 7.5 7 33 column shows the weight 7.5 to 8.5 8 29 interval, the second column 8.5 to 9.5 9 17 the average weight within 9.5 to 10.5 10 8 each weight interval, and 10.5 to 11.5 11 4 the third column the TOTAL 125 number of newborn infants at each weight. (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?
6. 6. 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
7. 7. z-scores 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
8. 8. 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?
9. 9. 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?
10. 10. 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 C 15 R 45 D 21 S 50 taste appeal. Each E 28 T 70 consumer group used a different rating HOMEWORK system. Use z-scores to determine which cereal has the higher taste appeal rating.
11. 11. 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 HOMEWORK (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?
12. 12. 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 HOMEWORK package of raisins?
13. 13. 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?