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Exercise: Empirical Rule. 1. Use the empirical rule to answer the following: ▫ Monthly maintenance costs are distributed normally with a µ=$250 and σ=$50.
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Use the empirical rule to answer the following: ! Monthly maintenance costs are distributed normally with a μ=$250 and σ=$ " 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) " 2)
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 = The standard score is positive for data values above the mean and negative for data values below the mean. data value – mean standard deviation
! 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).
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.
What percent of observations are below a standard score of z = 0? Ans: 50% A standard score of 0 is at the 50 th percentile. 50% shaded
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% + 94.52%) 2 Or 93.92%, which is the 93. nd percentile. 94% shaded
! 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.
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! 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?
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 16.
! Example : The Stanford-Binet IQ test is normally distributed with a mean of 100 and standard deviation of 16. Let X be an IQ score of a person. Short-hand notation: X ~ N(μ=100,σ=16) Normal 19 The 2 parameters needed to define a normal distribution. “is distributed”
" 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=16), what is P(X ≤ 90)? X is a data value (or IQ score in this case). We will convert it to a z-score…
