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Putting Statistics to Work

Discussion Paragraph 6B

1 web26. Web Data Sets

1 world27. Ranges in the News28. Summarizing a News Data Set29. Range Rule in the News

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Unit 6C

The Normal Distribution

6-C

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The Normal Distribution

The normal distribution is a symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution.

6-CThe Normal ShapeCN (1)

Look on p.392 Figure 6.14a shows a famous data set of the

chest sizes of 5738 Scottish Militiamen collected in about 1846.

Figure 6.14b Shows the distribution of the population densities of the 50 states.

1. Which distribution appears to be normal? Explain.

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6-C

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1. Most data values are clustered near the mean, giving the distribution a well-defined single peak.

2. Data values are spread evenly around the mean, making the distribution symmetric.

3. Larger deviations from the mean are increasingly rare, producing the tapering tails of the distribution.

4. Individual data values result from a combination of many different factors.

A data set satisfying the following criteria is likely to have a nearly normal distribution.

Conditions for a Normal Distribution

6-CIs It a Normal Distribution?CN (2a-b)

2. Which of the following variables would you expect to have a normal or nearly normal distribution.

a. Scores on a very easy test b. Shoe sizes of a random sample of adult

women

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6-C

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The 68-95-99.7 Rule for a Normal Distribution

6-CSAT ScoresCN (3)

Each test that makes up the SAT is designed so that its scores ar normally distributed with a mean of 500 and a standard deviation of 100.

3. Interpret this statement according to the 68-95-99.7 rule

Copyright © 2011 Pearson Education, Inc. Slide 6-9

6-CDetecting CounterfeitsCN (4)

Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal US quarters are normally distributed with a mean of 5.67 grams and a standard deviation of .0700 gram.

If a vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what percentage of legal quarters will be rejected by the machine?

Copyright © 2011 Pearson Education, Inc. Slide 6-10

6-CNormal Auto PricesCN (5a-c)

A survey finds that the prices paid for two-year-old Ford Fusion cars are normally distributed with a mean of $10,500 and a standard deviation of $500.

5. Consider a sample of 10,000 people who bought two-year-old Ford Fusions.

a. How many people paid between $10,000 and $11,000?

b. How many paid less than $10,000? c. How many paid more than $12,000?Copyright © 2011 Pearson Education, Inc. Slide 6-11

6-C

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Standard Scores

The number of standard deviations that a data value lies above or below the mean is called its standard score (or z-score), defined by

deviation standard

meanvalue datascore standard

z

Data Valueabove the meanbelow the mean

Standard Scorepositivenegative

→→

6-C

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Standard Scores

deviation standard

meanvalue data z

Example: If the mean were 21 with a standard deviation of 4.7 for scores on a nationwide test, find the z-score for a 30. What does this mean?

This means that a test score of 30 would be about 1.91 standard deviations above the mean of 21.

91.17.4

2130

6-CStandard IQ ScoresCN (6a-b)

6. The Standford-Binet IQ test is designed so that scores are normally distributed with a mean of 100 and a standard deviation of 16.

Find the standard scores for IQ scores of a)95 and b)25

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6-C

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The nth percentile of a data set is the smallest value in the set with the property that n% of the data are less than or equal to it.

A data value that lies between two percentiles is said to lie in the lower percentile.

Standard Scores and Percentiles

6-C

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Standard Scores and Percentiles

6-CCholesterol LevelsCN (7)

7. Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.

a. In what percentile is a man with cholesterol level of 190?

b. What cholesterol level corresponds to the 90th percentile, the level at which treatment may be necessary?

Copyright © 2011 Pearson Education, Inc. Slide 6-17

6-CWomen in the ArmyCN (8)

The heights of American women aged 18 to 24 are normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches. In order to serve in the US Army, women must be between 58 inches and 80 inches tall.

8. What percentage of women are ineligible to serve based on their height?

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6-CQuick QuizCN (9)

9. Please go through and answer the 10 multiple choice questions on p.398

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6-CHomework 6Cp.398

Discussion Paragraph for 6B p.398:1-10 1 web

48. SAT scores 49. Data and Story Library 50. Normal Distribution Demonstration

1 world Normal Distributions Non-Normal Distributions

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