Normal Distributions: Finding Values Larson/Farber 4th ed1

Section 5.3 Objectives Find a z-score given the area under the normal curve Transform a z-score to an x-value Find a specific data value of a normal distribution given the probability Larson/Farber 4th ed2

Finding values Given a Probability In section 5.2 we were given a normally distributed random variable x and we were asked to find a probability. In this section, we will be given a probability and we will be asked to find the value of the random variable x. Larson/Farber 4th ed3 xz probability

Example: Finding a z-Score Given an Area Find the z-score that corresponds to a cumulative area of Larson/Farber 4th ed4 z 0 z Solution:

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. Larson/Farber 4th ed5 The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is

Example: Finding a z-Score Given an Area Find the z-score that has 10.75% of the distribution’s area to its right. Larson/Farber 4th ed6 z0 z Solution: 1 – = Because the area to the right is , the cumulative area is

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. Larson/Farber 4th ed7 The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is 1.24.

Example: Finding a z-Score Given a Percentile Find the z-score that corresponds to P 5. Larson/Farber 4th ed8 Solution: The z-score that corresponds to P 5 is the same z-score that corresponds to an area of The areas closest to 0.05 in the table are (z = -1.65) and (z = -1.64). Because 0.05 is halfway between the two areas in the table, use the z-score that is halfway between and The z-score is z 0 z 0.05

Transforming a z-Score to an x- Score To transform a standard z-score to a data value x in a given population, use the formula x = μ + zσ Larson/Farber 4th ed9

Example: Finding an x-Value The speeds of vehicles along a stretch of highway are normally distributed, with a mean of 67 miles per hour and a standard deviation of 4 miles per hour. Find the speeds x corresponding to z-sores of 1.96, -2.33, and 0. Larson/Farber 4th ed10 Solution: Use the formula x = μ + zσ z = 1.96:x = (4) = miles per hour z = -2.33:x = 67 + (-2.33)(4) = miles per hour z = 0:x = (4) = 67 miles per hour Notice mph is above the mean, mph is below the mean, and 67 mph is equal to the mean.

Example: Finding a Specific Data Value Scores for a civil service exam are normally distributed, with a mean of 75 and a standard deviation of 6.5. To be eligible for civil service employment, you must score in the top 5%. What is the lowest score you can earn and still be eligible for employment? Larson/Farber 4th ed11 ? 0 z 5% ? 75 x Solution: 1 – 0.05 = 0.95 An exam score in the top 5% is any score above the 95 th percentile. Find the z- score that corresponds to a cumulative area of 0.95.

Solution: Finding a Specific Data Value From the Standard Normal Table, the areas closest to 0.95 are (z = 1.64) and (z = 1.65). Because 0.95 is halfway between the two areas in the table, use the z-score that is halfway between 1.64 and That is, z = Larson/Farber 4th ed z 5% ? 75 x

Solution: Finding a Specific Data Value Using the equation x = μ + zσ x = (6.5) ≈ Larson/Farber 4th ed z 5% x The lowest score you can earn and still be eligible for employment is 86.

Section 5.3 Summary Found a z-score given the area under the normal curve Transformed a z-score to an x-value Found a specific data value of a normal distribution given the probability Larson/Farber 4th ed14