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# Applied 40S April 8, 2009

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Working with z-scores: the three "meanings" of the area under the Normal Curve.

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### Applied 40S April 8, 2009

1. 1. Working the scores The letter Z by ﬂickr user Admit One
2. 2. ShadeNorm(Lo z, Hi z) [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
3. 3. Properties of a Normal Distribution The 68-95-99 Rule Generally speaking, approximately: • 68% of all the data in a normal distribution lie within the 1 standard deviation of the mean, • 95% of all the data lie within 2 standard deviations of the mean, and • 99.7% of all the data lie within 3 standard deviations of the mean.
4. 4. Case 1(a): Calculate the Percentage of Scores Between Two Given Scores HOMEWORK 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.
5. 5. HOMEWORK Find the percent of z-scores in a standard normal distribution that are: (b) above z = -2.35 (a) below z = 0.52 (d) between z = 0.55 and z = 0.15 (c) between z = -1.11 and z = 0.92
6. 6. HOMEWORK Find the z-score if the area under a standard normal curve: (a) to the left of z is 0.812 (b) to the right of z is 0.305 (c)to the right of z is 0.785
7. 7. HOMEWORK Determine the values for z and x.
8. 8. The 3 meanings of a Shaded Normal Curve area zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve)
9. 9. The 3 meanings of a Shaded Normal Curve % zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve) • a percentage (the percentage of all values in a data set that lie between two particular z-scores)
10. 10. The 3 meanings of a Shaded Normal Curve P(E) zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve) • a percentage (the percentage of all values in a data set that lie between two particular z-scores) • a probability (the probability that a particular z-score falls between two given z-scores)
11. 11. The 3 meanings of a Shaded Normal Curve area % P(E) zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve) • a percentage (the percentage of all values in a data set that lie between two particular z-scores) • a probability (the probability that a particular z-score falls between two given z-scores)
12. 12. Case 1(c): Calculate the Number of Scores 1200 light bulbs were tested for the number of hours of life. The mean life was 640 hours with a standard deviation of 50 hours. Assume that quot;life in hoursquot; of light bulbs is normally distributed. (a) What percent of light bulbs lasted between 600 and 700 hours? (b) How many light bulbs should be expected to last between 600 and 700 hours?
13. 13. Case 1(b): Calculate the Percentage of Passing 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 passing mark if they must get 60 percent or better to pass? Assume that the marks are normally distributed. HOMEWORK
14. 14. Case 2(a): Calculate the Percentage of Scores Between Two Z-Scores HOMEWORK (Case 2) If we know two z-scores of a standard normal distribution, we can ﬁnd the percentage of scores that lie between them. The procedure is similar to that used in the previous examples. Sample question(s):What percent of scores lie between z = 0.87 and z = 2.57? OR What is the probability that a score will fall between z = 0.87 and z = 2.57? OR Find the area between z = 0.87 and z = 2.57 in a standard normal distribution.
15. 15. Case 2(b): Calculate the Percentage of Scores Between Two Z-Scores HOMEWORK Find the probability of getting a z-score less than 0.75 in a standard normal distribution.
16. 16. Case 3(b): Find the Z-Value that Corresponds to a Given Probability HOMEWORK What is the z-score if the probability of getting less than this z-score is 0.750?
17. 17. Case 3(a): Find the Z-Score that Corresponds to a Given Probability HOMEWORK If we know the probability of an event, we can ﬁnd the z-score that corresponds to this probability. This is the reverse of what we did in Case 2. Sample question: What is the z-score if the probability of getting more than this z-score is 0.350?