Exercise: Empirical Rule. Answer: " 1) STAT1010 Standard Scores. 5.2 Properties of the Normal Distribution

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1 Exercise: Empirical Rule Use the empirical rule to answer the following:! Monthly maintenance costs are distributed normally with a µ=$250 and σ=$50 " 1) What percent of months have maintenance costs in the range of $200 to $300? " 2) What is the chance (i.e. probability) that a randomly chosen month has a maintenance cost of $150 or less? 1 Answer: " 1) " 2) Properties of the Normal Distribution part 2!! Finding percentiles when you can t use the empirical rule (when the data value is something other than 1, 2 or 3 standard deviations from the mean) 3 1

2 Normal Percentiles! What if we re 1.5 standard deviations up from the mean? How do we compute such a percentile?! Solution: 4 The number of standard deviations a data value lies above or below the mean is called its standard score (or z-score), defined by z = standard score = data value mean standard deviation The standard score is positive for data values above the mean and negative for data values below the mean. 5! Example (continued): The Stanford-Binet IQ test is scaled so that scores have a mean of 100 and a standard deviation of. Find the standard scores for IQs of 85, 100, and standard score for IQ of 85: z = = standard score for IQ of 100: z = = standard score for IQ of 125: z = = 1.56 An IQ of 85 is 0.94 standard deviations below the mean. 6 2

3 ! What percentile are the following IQ scores? " 85 (standard score: z = -0.94) " 100 (standard score: z = 0) " 125 (standard score: z = 1.56)! We can t use the empirical rule here.! We ll have to use a table to find the percentages (Appendix A in our book). 7 A subset of Appendix A (shown here) is provided in Section 5.2 in the book. This table shows the percentage of observations below any given standard score. 8 What percent of observations are below a standard score of z = -0.94? The closest standard score is and 17.11% of the observations are below a standard score of % shaded 9 3

4 What percent of observations are below a standard score of z = 0? Ans: 50% 50% shaded A standard score of 0 is at the 50 th percentile. 10 What percent of observations are below a standard score of z = 1.56? This standard score lies between 1.5 and 1.6 on the table. We can approximate the percentile for this standard score as (93.32% %) 2 Or 93.92%, which is the nd percentile. 94% shaded 11! What percentile are the following IQ scores? " IQ of 85 (z=-0.94) is at the 17 th percentile. " IQ of 100 (z=0) is at the 50 th percentile " IQ of 125 (z=1.56) is at the 94 th percentile. " Recall that negative z-scores are below the mean and positive z-scores are above the mean.! Thus, we can get the percentiles even though we re not exactly 1, 2 or 3 standard deviations from the mean. 12 4

5 ! We will use Appendix A from the book (a subset of that table was shown above) to compute percentiles and probabilities because it has finer resolution (more decimals).! THIS TAKES PRACTICE!! Active work: See worksheet on normal curve scores. 13 More Exercises: z-scores! Assume you have a normal distribution. Use the z-score table in Appendix A to answer: " 1) What percent of observations lie below a z- score of 0? " 2) What percent of observations lie below a z- score of 1.72? 14 More Exercises: z-scores " 3) What percent of observations fall BETWEEN z-scores of 0 and 1.72? 15 5

6 5.2 Properties of the Normal Distribution part 3! Connecting z-scores to probabilities.! Example: The Stanford-Binet IQ test is normally distributed and scaled so that scores have a mean of 100 and a standard deviation of. 17! Example: The Stanford-Binet IQ test is normally distributed and scaled so that scores have a mean of 100 and a standard deviation of. " If you draw someone at random, what is the probability that they have an IQ score of 90 or less? " To answer this, we just need to know what percent of IQ scores are at 90 or lower. 18 6

7 ! Example: The Stanford-Binet IQ test is normally distributed with a mean of 100 and standard deviation of. Let X be an IQ score of a person. Short-hand notation: The 2 parameters needed to define a normal distribution. X ~ N(µ=100,σ=) is distributed Normal 19 " If you draw someone at random, what is the probability that they have an IQ score of 90 or less? " We need to answer: When X ~ N(µ=100,σ=), what is P(X 90)? X is a data value (or IQ score in this case). We will convert it to a z-score 20 data value mean z = standard score = standard deviation " P(X 90) = P( X µ ) σ = P(Z 10/) = P(Z 0.63) =

8 data value mean z = standard score = standard deviation " P(X 90) = P(Z 0.63) = Looked up on z-table An IQ score of 90 has a z-score of " The probability of randomly drawing someone with an IQ score of 90 or lower is QUICK-CHECK: The Empirical Rule tells me the percent that is below an IQ of 90 has to be between% (to the left of 1σ below the mean) and 50% (to the left of the mean itself). IQ 90 So, 26.43% is totally in-line with my Empirical Rule information because being 0.63 standard deviations is between 1 and 0 standard deviations down from the mean. 23 Using new notation, exercise 1:! Let X ~ N(µ=40,σ=5). Find P(X < 51): 24 8

9 Using new notation, exercise 2:! Suppose bowling scores are normally distributed with a mean of 186 and a standard deviation of 30. Find the percentage of games with a score of 120 or HIGHER. 25 9