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Understanding Normal Distributions and Z-Scores in Statistics

karl avatar
karl
27 diapositives · 423 Views · Jul 17, 2014

This text explores the concept of normal distributions, particularly focusing on how standardization through z-scores enables comparison of data. It details real-world examples, including the heights of young women and cholesterol levels in 14-year-old boys. Steps are provided for standardizing values, finding corresponding areas under the standard normal curve using z-tables, and interpreting results in context. Through practical scenarios, we see how statistical methods are applied to understand the distribution and proportions in normal datasets.

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Understanding Normal Distributions and Z-Scores in Statistics
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  1. Standard Normal Calculations Section 2.2

  2. Normal Distributions • Can be compared if we measure in units of size σabout the mean µ as center. • Changing these units is called standardizing.

  3. Standardizing and z-scores • If x is an observation from a distribution that has mean µand standard deviation σ, the standardized value of x is A standardized value is often called a z-score.

  4. Heights of Young Women • The heights of young women are approximately normal with µ = 64.5 inches and σ= 2.5 inches. The standardized height is

  5. Heights of Young Women • 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.

  6. Heights of Young Women • A woman who is 5 feet (60 inches) tall has a standardized height or 1.8 standard deviations less than the mean.

  7. Standard Normal Distribution • The normal distribution N(0,1) with mean 0 and standard deviation 1. If a variable x has any normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable has the standard normal distribution.

  8. The Standard Normal 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.

  9. Using the z Table • 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.

  10. Using the z Table • 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.

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

  12. Find the proportion of observations from the standard normal distribution that are greater than -2.15. • z = 0.0158 • Remember, Table A gives us what is less than a z-score. 1 – 0.0158 = .9842

  13. Steps for Finding Normal Distribution 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.

  14. Cholesterol Problem • 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

  15. Cholesterol Problem • 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

  16. Cholesterol Problem • 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%

  17. Cholesterol Problem • Step 4: Write your conclusion in the context of the problem. Only about 1% of boys have high cholesterol.

  18. Working with an Interval • 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

  19. Working with an Interval • 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

  20. Working with an Interval • 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.

  21. Finding a Value given a Proportion • 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

  22. Finding a Value given a Proportion • 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.

  23. Finding a Value given a Proportion • 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?

  24. Finding a Value given a Proportion • 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? • z-score = 1.28 Unstandardize:

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