stats sb notes 5.3 completed.notebook
TRANSCRIPT
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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ChapterNormal Probability Distributions
5
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Chapter Outline
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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Section 5.3Normal Distributions: Finding Values
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Section 5.3 Objectives• How to find a zscore given the area under the normal curve
• How to transform a zscore to an xvalue
• How to find a specific data value of a normal distribution given the probability
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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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.
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Example: Finding a zScore Given an Area
Find the zscore that corresponds to a cumulative area of 0.3632.
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Solution: Finding a zScore Given an Area
• Locate 0.3632 in the body of the Standard Normal Table.
• The values at the beginning of the corresponding row and at the top of the column give the zscore.
The zscore is −0.35.
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Example: Finding a zScore Given an Area
Find the zscore that has 10.75% of the distribution’s area to its right.
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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Solution: Finding a zScore Given an Area
• Locate 0.8925 in the body of the Standard Normal Table.
• The values at the beginning of the corresponding row and at the top of the column give the zscore.
The zscore is 1.24.
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Try It Yourself, pg 253.
1. Find the z-score that has 96.16% of the distribution's area to its right.
2. Find the z-score for which 95% of the distribution's area lies between -z and z.
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Example: Finding a zScore Given a Percentile
Find the zscore that corresponds to:
1.P5 2.P50 3.P90
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Try It Yourself 2, pg 253
Find the z-score that corresponds to each percentile.
1. P10 2. P20 3. P99
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Transforming a zScore to an xScore
To transform a standard zscore to a data value x in a given population, use the formula
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Example: Finding an xValueA veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to zscores of 1.96, 0.44, and 0.
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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Try It Yourself 3, pg 254.
A veterinarian records the weights of dogs treated at a clinic. The weights are normally distributed, with a mean of 52 pounds and a standard deviation of 15 pounds. Find the weights x corresponding to z-scores of -2.33, 3.10, and 0.58. Interpret your results.
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Example: Finding a Specific Data Value
Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency?
An exam score in the top 10% is any score above the 90th percentile. Find the zscore that corresponds to a cumulative area of 0.9.
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Solution: Finding a Specific Data Value
From the Standard Normal Table, the area closest to 0.9 is 0.8997. So the zscore that corresponds to an area of 0.9 is z = 1.28.
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Solution: Finding a Specific Data Value
Using the equation x = μ + zσ
x = 50 + 1.28(10) ≈ 62.8
The lowest score you can earn and still be eligible to be hired by the agency is about 63.
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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Try It Yourself 4, pg 255.
A researcher tests the braking distances of several cars. The breaking distance from 60 miles per hour to a complete stop on dry pavement is measured in feet. The braking distances of a sample of cars are normally distributed, with a mean of 129 feet and a standard deviation of 5.18 feet. What is longest braking distance one of these cars could have and still be in the bottom 1%?
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Example 5 Finding a Specific Data Value
In a randomly selected sample of women ages 20-34, the mean total cholesterol level is 181 milligrams per deciliter with a standard deviation of 37.6 milligrams per deciliter. Assume the total cholesterol levels are normally distributed. Find the the highest total cholesterol level a woman in this 20-34 age group can have and still be in the bottom 1%?
Stats SB Notes 5.3 Completed.notebook March 21, 2017
Mar 108:29 AM
Try It Yourself 5, pg 256.
The lengths of time employees have worked at a corporation are normally distributed, with a mean of 11.2 years and a standard deviation of 2.1 years. In a company cutback, the lowest 10% in seniority are laid off. What is the maximum length of time an employee could have worked and still be laid off?
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Section 5.3 Summary• Found a zscore given the area under the normal curve
• Transformed a zscore to an xvalue
• Found a specific data value of a normal distribution given the probability
Stats SB Notes 5.3 Completed.notebook March 21, 2017
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Stats HW Section 5.3
pg 257, 2-42 Evens