2. Putting Statistics to Work
Discussion Paragraph 6B
1 web
26. Web Data Sets
1 world
27. Ranges in the News
28. Summarizing a News Data Set
29. Range Rule in the News
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3. The Normal Distribution
Unit 6C
The Normal Distribution
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4. 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.
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5. The Normal Shape CN (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. Conditions for a Normal Distribution
A data set satisfying the following criteria is likely to have a nearly normal distribution.
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.
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7. Is 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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8. The 68-95-99.7 Rule for a Normal Distribution
About 68.3% of the data points fall within 1 standard deviation of the mean. About 95.4% of the data points fall within 2 standard deviation points of the mean. About 99.7% of the data points fall within 3 standard deviation points of the mean. This is a wonderful opportunity to talk about what makes a distribution normal. The authors suggest in the reading that when any quantity is a result of many factors, such as genetic or environmental, there is a definite tendency for normal distribution. Apply the rule to male or female heights of your own class or some other quantitative variable that results from many factors to check out that the percentages are reasonable.
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9. SAT Scores CN (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 rule
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10. Detecting Counterfeits CN (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 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?
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11. Normal Auto Prices CN (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?
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12. 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
Stress with the students that the formula should not be a mystery to them but rather it simply tells us how many standard deviations the data value is away from the mean.
Data Value
above the mean
below the mean →
Standard Score
positive
negative
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13. Standard Scores
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.
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14. Standard IQ Scores CN (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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15. Standard Scores and Percentiles
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.
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16. Standard Scores and Percentiles
Work through some simple concrete examples to demonstrate how the table works. Try different scenarios that require both adding and subtracting percentages. It may be helpful to emphasize the importance of sketching out a normal curve in the problem-solving process in order to visual what it is we are actually doing.
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17. Cholesterol Levels CN (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?
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18. Women in the Army CN (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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19. Quick Quiz CN (9)
9. Please go through and answer the 10 multiple choice questions on p.398
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20. Homework 6C p.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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