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The Standard Normal Distribution

The Standard Normal Distribution. Z-score Empirical Rule Normal Distribution. Z-score: Important notes. Using the Z-Score formula to standardized values Drawing the Normal Density curve using the Empirical Rule Locating your Z distribution using the Z-table.

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The Standard Normal Distribution

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  1. The Standard Normal Distribution • Z-score • Empirical Rule • Normal Distribution

  2. Z-score: Important notes • Using the Z-Score formula to standardized values • Drawing the Normal Density curve using the Empirical Rule • Locating your Z distribution using the Z-table

  3. 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 Biology test. Given the test’s Normal distribution of N(130, 34), she wants to find how well she did relative to her classmates’ performance on the test. Find the proportion of Sprite’s sugar content compared to the other soft drink in this data set.

  4. Answers: 6. Sprite is at the 63rd percentile on the popular soft drinks which gives high sugar content in a bottle of soda (1).5596 (2) .4404 (3) .0039 (4) .0796 (5.) Martha is only on the 27th percentile on that Biology quiz, which means only 27% of her classmates has the same or lower score compared to her score.

  5. Constructing Normal probability plot using your calculator

  6. Is cholesterol a problem for young boys? 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?

  7. 1. Draw the Normal Curve Proportion under the normal curve Cholesterol levels for 14-year-old boys who may require medical attention.

  8. 2. Standardized the value and sketch the Standard Normal Curve

  9. 3. Use the Table to find the 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%.

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

  11. Solving Problems Involving Normal Distributions 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.

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