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# Applied Math 40S March 20, 2008

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Introduction to working with z-scores on standard and modified Normal curves.

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### Applied Math 40S March 20, 2008

1. 1. Working with z-scores z by Shamana ©
2. 2. Professor Adams has 140 students who wrote a statistics test. If the marks are approximately normally distributed: • how many students should have a 'B' mark (i.e., 70 to 79 percent)? • how many students should have failed (i.e., less than 50 percent)? • how high must she set the mark for an 'A' if she wants 5 percent of the students to get an A? • how high must she set the passing mark if she wants only the top 75 percent of the marks to be passing marks?
3. 3. A group of 60 Applied Math students obtained the following scores on the final examination last year. 60 64 59 56 48 39 65 64 63 53 92 64 69 72 77 51 75 53 76 79 90 48 39 76 46 27 23 55 65 82 76 54 72 27 72 48 67 40 85 36 96 68 72 81 36 55 51 94 69 57 68 92 39 48 37 63 82 43 72 37 Draw a suitable histogram that has 10 bars.
4. 4. Working with the Normal Distribution Example: The graph at right represents the marks of a large number of students where the mean mark is 69.3 percent and the standard deviation is 7 percent, and the distribution of marks is approximately normal. 62.3% 69.3% 76.3%
5. 5. Working with the Normal Distribution Example: The graph at right represents the marks of a large number of students where the mean mark is 69.3 percent and the standard deviation is 7 percent, and the distribution of marks is approximately normal. We know that if the marks are approximately normally distributed, then approximately: • 68 percent of the marks are between 62.3 percent and 76.3 percent (i.e., µ ± 1σ) • 34% of the marks are between 69.3 percent and 76.3 percent (i.e., between µ and µ + 1σ) • 50% of the marks are below 69.3 percent (i.e., less than µ) • 16% of the marks are above 76.3 percent (i.e., greater than µ + 1σ)
6. 6. RECALL 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?
7. 7. ShadeNorm(Lo z, Hi z) DICTIONARY [shades area under std. normal curve] σ ShadeNorm(Lo value, Hi value, mean, std. dev.) [shades area under modified normal curve] µ-5σ σ µ+5σ
8. 8. Case 1(a): Calculate the Percentage of Scores Between Two Given Scores The mean mark for a large number of students is 69.3 percent with a standard deviation of 7 percent. What percent of the students have a 'B' mark (i.e., 70 percent to 79 percent)? Assume that the marks are normally distributed.