The Standard Normal The Standard Normal DistributionDistribution

Z-scoreZ-score

Empirical RuleEmpirical Rule

Normal DistributionNormal Distribution

Z-score: Important Z-score: Important notesnotes

Using the Z-Score formula to Using the Z-Score formula to standardized valuesstandardized values

Drawing the Normal Density Drawing the Normal Density curve using the Empirical Rulecurve using the Empirical Rule

Locating your Z distribution Locating your Z distribution using the Z-tableusing the Z-table

Find the following proportions of the following:Find the following proportions of the following:(1) z < .85(2) z > .85(3) z > 2.66(4) −.1 < z < .1

(5.) Martha got 109 points on her (5.) Martha got 109 points on her Biology test. Given the testBiology test. Given the test’’s s Normal distribution of N(130, 34), Normal distribution of N(130, 34), she wants to find how well she did she wants to find how well she did relative to her classmatesrelative to her classmates’’ performance on the test.performance on the test.

Find the proportion of SpriteFind the proportion of Sprite’’s sugar content s sugar content compared to the other soft drink in this data compared to the other soft drink in this data

set.set.

(1).5596(2) .4404(3) .0039(4) .0796

(5.) Martha is only on the 27th (5.) Martha is only on the 27th percentile on that Biology quiz, percentile on that Biology quiz, which means only 27% of her which means only 27% of her classmates has the same or lower classmates has the same or lower score compared to her score.score compared to her score.

6. Sprite is at the 63rd percentile on the 6. Sprite is at the 63rd percentile on the popular soft drinks which gives high sugar popular soft drinks which gives high sugar

content in a bottle of sodacontent in a bottle of soda

Answers:Answers:

Constructing Constructing Normal probability Normal probability plotplot using your using your calculatorcalculator

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?

Is cholesterol a problem for young boys?

1. Draw the Normal 1. Draw the Normal CurveCurve

Cholesterol levels for 14-year-old boys who may require medical attention.

Proportion Proportion under the under the

normal curvenormal curve

Proportion Proportion under the under the

normal curvenormal curve

2. Standardized the value and 2. Standardized the value and sketch the Standard Normal sketch the Standard Normal CurveCurve

3. Use the Table to find 3. Use the Table to find the Value of Zthe Value of Z

From Table A, we see that the proportion of observations less than 2.33 is 0.9901. About 99% of boys have cholesterol levels less than 240. The area to the right of 2.33 is therefore 1 − 0.9901 = 0.0099. This is about 0.01, or 1%.

4. Write your conclusion 4. Write your conclusion in the context of the in the context of the problem.problem.

Step 1: State the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve.

Step 2: Standardize and draw a picture. Standardize x to restate the problem in terms of a standard Normal variable z. Draw a picture to show the area of interest under the standard Normal curve.

Step 3: Use the table.

Step 4: Conclusion. Write your conclusion in the context of the problem.

Solving Problems Involving Normal Distributions