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S TANDARD N ORMAL C ALCULATIONS Section 2.2

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N ORMAL D ISTRIBUTIONS Can be compared if we measure in units of size σ about the mean µ as center. Changing these units is called standardizing.

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S TANDARDIZING AND Z - SCORES A standardized value is often called a z -score.

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H EIGHTS OF Y OUNG W OMEN The heights of young women are approximately normal with µ = 64.5 inches and σ = 2.5 inches. The standardized height is

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H EIGHTS OF Y OUNG W OMEN A woman’s standardized height is the number of standard deviations by which her height differs from the mean height of all women. For example, a woman who is 68 inches tall has a standardized height or 1.4 standard deviations above the mean.

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H EIGHTS OF Y OUNG W OMEN A woman who is 5 feet (60 inches) tall has a standardized height or 1.8 standard deviations less than the mean.

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S TANDARD N ORMAL D ISTRIBUTION The normal distribution N (0,1) with mean 0 and standard deviation 1.

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T HE S TANDARD N ORMAL TABLE Table A (front of your book) is a table of areas under the standard normal curve. The table entry for each value z is the area under the curve to the left of z.

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U SING THE Z T ABLE Back to our example of women 68 inches or less. We had a z -score of 1.4. To find the proportion of observations from the standard normal distribution that are less 1.4, locate 1.4 in Table A.

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U SING THE Z T ABLE Back to our example of women 68 inches or less. We had a z -score of 1.4. To find the proportion of observations from the standard normal distribution that are less 1.4, locate 1.4 in Table A. What does this mean? About 91.92% of young women are 68 inches or shorter.

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Find the proportion of observations from the standard normal distribution that are greater than -2.15.

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Proportion = 0.0158 Remember, Table A gives us what is less than a z -score. 1 – 0.0158 =.9842

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S TEPS FOR F INDING N ORMAL D ISTRIBUTION Step 1: State the problem in terms of the observed variable x. Step 2: Standardize x to restate the problem in terms of a standard normal curve. Draw a picture of the distribution and shade the area of interest under the curve. Step 3: Find the required area under the standard normal curve, using Table A and the fact that the total area under the curve is 1. Step 4: Write your conclusion in the context of the problem.

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C HOLESTEROL P ROBLEM The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14-year old boys, the mean is µ = 170 milligrams of cholesterol per deciliter of blood (mg/dl) and the standard deviation is σ = 30 mg/dl. Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol? Step 1: State the Problem. Level of cholesterol = x x has the N(170,30) distribution Want the proportion of boys with cholesterol level x > 240

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C HOLESTEROL P ROBLEM The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14-year old boys, the mean is µ = 170 milligrams of cholesterol per deciliter of blood (mg/dl) and the standard deviation is σ = 30 mg/dl. Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol? Step 2: Standardize x and draw a picture

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C HOLESTEROL P ROBLEM Step 3: Use the Table (z > 2.33) 0.9901 is the proportion of observations less than 2.33. 1 – 0.9901 = 0.0099 About 0.01 or 1%

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C HOLESTEROL P ROBLEM Step 4: Write your conclusion in the context of the problem. Only about 1% of boys have high cholesterol.

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W ORKING WITH AN I NTERVAL What percent of 14-year-old boys have blood cholesterol between 170 and 240 mg/dl? Step 1: State the problem We want the proportion of boys with Step 2: Standardize and draw a picture

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W ORKING WITH AN I NTERVAL What percent of 14-year-old boys have blood cholesterol between 170 and 240 mg/dl? Step 3: Use the table z < 0 0.5000 z < 2.33 0.9901 0 < z < 2.33 0.9901 – 0.5000 = 0.4901

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W ORKING WITH AN I NTERVAL What percent of 14-year-old boys have blood cholesterol between 170 and 240 mg/dl? Step 4: State your conclusion in context About 49% of boys have cholesterol levels between 170 and 240 mg/dl.

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F INDING A V ALUE GIVEN A P ROPORTION Scores on the SAT Verbal test in recent years follow approximately the N (505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT? Find the SAT score x with 0.1 to its right under the normal curve. (Same as finding an SAT score x with 0.9 to its left ) µ = 505, σ = 110

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F INDING A V ALUE GIVEN A P ROPORTION Scores on the SAT Verbal test in recent years follow approximately the N (505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT? USE THE TABLE!!! Go backwards.

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F INDING A V ALUE GIVEN A P ROPORTION Scores on the SAT Verbal test in recent years follow approximately the N (505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

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F INDING A V ALUE GIVEN A P ROPORTION

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Homework 2.19 p. 95 2.21, 2.22, 2.23 p. 103

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