Section 5.5 Z Scores soln.notebook. December 07, Section 5.5 Z Scores

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1 Section 5.5 Z Scores 1

2 Warm up/review: The Normal Distribution Curve Given that the average adult in North America has a mean mass of 72 kg, with a standard deviation of 14 kg. a) How many standard deviations are the weights 58 kg and 86 kg away from the mean? b) What percentage of the population has weights between 58 and 86 kg? c) How many standard deviations are the weights 65 kg and 79 kg away from the mean? d) Given the rule, can we answer the percentage of the population that has weights between 65 kg and 79 kg? 2

3 In order to calculate the percentage under a normal curve for any given interval, statisticians have devised the z score table. The z score table represents the percentage underneath that is less than the given number of standard deviations away from the mean. Recall: Facts about Normal Distribution Curves 3

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5 What is the point of finding a Z score? Then refer to the chart p at the back of your book. The z score will give you a percent for the area under the curve, less than or equal to the data value. 5

6 Example 1: Determine the following z scores for data point if the mean is 67 and the standard deviation of 8.5. A) 90 B) 50 C) 4 D) 100 6

7 Example 2: On the math placement test at Memorial University of Newfoundland, the mean score was 62 and the standard deviation was 11. If Mark s z score was 0.8, what was his actual exam mark? 7

8 Example 3: IQ tests are sometimes used to measure a person s intellectual capacity at a particular time. IQ scores are normally distributed, with a mean of 100 and standard deviation of 15. a) If a person scores 119 on an IQ test, how does this score compare with the scores of the general population? b) If this data reflects that of a population of people, approximately how many people have an IQ above 119? 8

9 Example 4: Draw and label a standard normal distribution and find the percentage of scores that would be: A) Below z = 2.25 B) Above z =

10 C) Below z = 1.78 D) Above z =

11 E) Between z = 2.1 and z = 1.2 F) Between z = 1 and z = 1 11

12 Example 5: Two students competed in a nationwide mathematics competition and received these scores. Anna: 70 Bruce: 80 a) If μ = 66 and σ = 10, find their z scores. b) What percent of the population did Anna score better than? c) What percent of the population, scored better than Bruce? 12

13 Example 6: The average life expectancy of a certain breed of cat was determined to be 12.2 years with a standard deviation of 1.3 years. What is the probability that a given cat will live less than 14 years? 13

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15 Example 7: Red candy hearts are packaged according to weight with a mean of 300 g and a standard deviation of 8 g. Packages with weights less than 290 g and more than 312 g are rejected by quality control workers. a) If packages are produced each day, how many packages would quality control expect to reject in a day? b) What advice would you give this company? 15

16 Example 8: Cars are undercoated as a protection against rust. A car dealer determines the mean life of protection is 65 months and the standard deviation is 4.5 months. a) What guarantee should the dealer give so that fewer than 15% of the customers will return their cars? b) The dealer creates a fund, based on the guarantee, from which refunds and repairs are made. It is estimated that about 2500 cars will be undercoated annually. The average repair on returned cars is about $165. How much money should be placed in the fund to cover customer returns? 16

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