The Normal Model. Copyright 2009 Pearson Education, Inc.

Transcription

1 The Normal Mol Copyright 2009 Pearson Education, Inc.

2 The trick in comparing very different-looking values is to use standard viations as our rulers. The standard viation tells us how the whole collection of values varies, so it s a natural ruler for comparing an individual to a group. As the most common measure of variation, the standard viation plays a crucial role in how we look at data. 2

3 We compare individual data values to their mean, relative to their standard viation using the following formula: z y y s We call the resulting values standardized values, noted as z. They can also be called z-scores. 3

4 a) Alex s score on a test was 84 points. The class average was 78 and the standard viation was 6 points. What was her z- score? a) The average score on a psych test was 65 points with a standard viation of 15 points. Joe s z-score was 2. How many points did he score? 4

5 z-scores measure the distance of each data value from the mean in standard viations. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. 5

6 Standardized values have been converted from their original units to the standard statistical unit of standard viations from the mean. Thus, we can compare values that are measured on different scales, with different units, or from different populations. 6

7 Standardizing data into z-scores shifts the data by subtracting the mean and rescales the values by dividing by their standard viation. Standardizing into z-scores does not change the shape of the distribution. Standardizing into z-scores changes the center by making the mean 0. Standardizing into z-scores changes the spread by making the standard viation 1. 7

8 Susan and Phil took math exams in separate classes. Susan scored 82 on her exam. Overall the stunt scores for the her class had a mean of 79 and a standard viation of 5. Phil scored 79 on his exam. Overall the stunt scores in his class had a mean of 70 and a standard viation of 12. Which stunt s overall performance was better? 8

9 A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. The larger a z-score is (negative or positive), the more unusual it is. 9

10 There is no universal standard for z-scores, but there is a mol that shows up over and over in Statistics. This mol is called the Normal mol (You may have heard of bell-shaped curves. ). Normal mols are appropriate for distributions whose shapes are unimodal and roughly symmetric. These distributions provi a measure of how extreme a z-score is. 10

11 The mean weight of a breed of dog is 45 pounds. Suppose that weight of all such animals can be scribed by a Normal mol with a standard viation of 5 pounds. How many standard viation from the mean would a dog weighing 38 pounds be? Which would be more unusual, a dog weighing 38 pounds or a dog weighing 55 pounds? 11

12 The mean weight of a watermelon is 68 ounces with a standard viation of 15 ounces. Suppose that weights of all such fruit can be scribes with a Normal mol. Watermelon buyers hope that watermelons will weigh at least 50 ounces. To see how much over or unr that goal the watermelons are, we could subtract 50 ounces from all the weights. What would the new mean and standard viation be? Suppose each watermelon sells for 65 cents per ounce. Find the mean and standard viation for the sale price of all the watermelons. 12

13 There is a Normal mol for every possible combination of mean and standard viation. We write Y~N (μ,σ) to represent a Normal mol with a mean of μ and a standard viation of σ. When we standardize Normal data, we still call the standardized value a z-score, and we write z y 13

14 Once we have standardized, we need only one mol: The N (0,1) mol is called the standard Normal mol (or the standard Normal distribution). Be careful don t use a Normal mol for just any data set, since standardizing does not change the shape of the distribution. 14

15 Normal mols give us an ia of how extreme a value is by telling us how likely it is to find one that far from the mean. We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the Normal mol 15

16 It turns out that in a Normal mol: about 68% of the values fall within one standard viation of the mean; about 95% of the values fall within two standard viations of the mean; and, about 99.7% (almost all!) of the values fall within three standard viations of the mean. 16

17 The following shows what the Rule tells us: 17

18 The EPA fuel economy estimates for automobile mols tested recently predicted a mean of 24.8 mpg and a standard viation of 6.2 mpg for highway driving. Assume that a Normal mol can be applied. A) Draw a mol for the auto fuel economy. Clearly label it, sowing what the Rule predicts about miles per gallon. B) In what interval would you expect the central 68% of the autos to be found? C) About what percent of autos should get more than 31 mpg? D) About what percent of cars should get between 31 and 37.2 mpg? E) Describe the gas mileage of the worst 2.5% of the cars. 18

19 When a data value doesn t fall exactly 1, 2, or 3 standard viations from the mean, we can look it up in a table of Normal percentiles. Or use technology Normcdf (lower value, higher value, μ, σ) 19

20 In a Standard Normal table. We have to convert our data to z-scores before using the table. The figure shows us how to find the area to the left when we have a z-score of 1.80: 20

21 The cholesterol levels of an adult can be scribed by a normal mol with a mean of 200 mg/dl and a standard viation of 15. What percent of adults do you expect to have cholesterol levels over 220 mg/dl? What percent of adults do you expect to have cholesterol levels between 175 and 185 mg/dl? Estimate the IQR of cholesterol levels. 21

22 Sometimes we start with areas and need to find the corresponding z-score or even the original data value. Example: What z-score represents the first quartile in a Normal mol? 22

23 In a Normal mol, what value(s) of z cut(s) off the region scribed? a) The highest 25% b) The highest 65% c) The lowest 65% d) The middle 90% Now we need the Inverse Normal Distribution, InvNorm (area to the left, μ, σ) 23

24 In the Normal mol for IQ scores N(100,16), what IQ Score bounds a) The highest 5% of all IQs? b) The lowest 30% of the IQs? c) The middle 80% of the IQs? 24

25 What percent of a standard Normal mol is found in each region? A) z > 2.1 B) z < 1.2 C) < z < 2.05 D) z >

26 Only 10% of babies have learned to walk by the age of 9 months and 85% of babies are walking by 14 months of age. If the age at which babies velop the ability to walk can be scribed by a normal mol, find the parameters (mean and standard viation). 26

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