Name:__________________________________ Date:_______________ Statistics

Measures of Relative Position NOTES 1. Suppose as student has taken two quizzes in a statistics course. On the first quiz the mean score was

32, the standard deviation was 8, and the student received a 44. The student obtained a 28 on the second quiz, for which the mean was 23 and the standard deviation was 3. If test scores are approximately normal, on which quiz did the student perform better relative to the rest of the class?

2. In 1993, a civil engineer graduating with a bachelor’s degree was offered on average $29,211.  A civil 

engineer graduating with a master’s degree earned $34,606.  Engineers graduating with bachelor’s degree can be represented by N($34286,$3636) and engineers graduating with master’s degrees can be represented by N($39278,$2731). Based on relative position, would you suggest someone to stay in school or “take the money and run?”

3. A married couple is employed by the same company. The husband works in a department for which

the mean hourly rate is $12.80 and the standard deviation is $1.20. His wife is employed in a department where the mean rate is $13.50 and the standard deviation is $1.80. Both department salaries are normally distributed. x Relative to their departments, which is better paid if the husband earns $14.60 and the wife earns

$15.75? x What percentile is the husband located in his department? x What percent of employees in the wife’s department earn better than her? x What would the wife need to earn to match her husband’s relative position? x If the husband wanted to earn in the 95th percentile, how much should he earn per hour?

4. A pizza parlor franchise specifies that the average (mean) amount of cheese on a large pizza should be

8 oz and the standard deviation only 0.5 oz. and normally distributed. An inspector picks out a large pizza at random in one of the pizza parlors and finds that it is made with 6.9 oz of cheese. If the amount of cheese is more than three standard deviations below the mean, the parlor will be in danger of losing its franchise. Should they be worried? What is the minimum amount of cheese that can be placed on a large pizza according to the franchise?

5. A study of elite distance runners found a mean body weight to be normally distributed with a mean of

63.1 kilograms (KG), with a standard deviation of 4.8 kg. x What percent of the runners have a body weight less than 60 kg? x What percent of the runners have a body weight greater than 70 kg? x What percent of the runners have a body weight between 55 and 75 kg? x A runner weighing 80 kg would be in what percentile?

6. Mobile homes are very tightly constructed for energy conservation. This may lead to a buildup of pollutants generated indoors.  The paper “A Survey of Nitrogen Dioxide Levels Inside Mobile Homes” discussed various aspects of NO2 concentration in these structures.

x In one sample of homes in the Los Angeles area, the mean NO2 concentration in kitchens during

the summer was 36.92 ppb, and the standard deviation was 11.32 and was approximately normal. What percentage of observations fall between 14.24 and 59.60?

x Inside what interval is it guaranteed that at least 89% of the concentration observations will lie? 7. Student cars at a particular high school have ages with a mean of 7.90 years and a standard deviation

of 3.67 years with the data resembling a bell curve. x What percent of students have a car between 6 and 10 years old? x What percent of the students have cars older than 12 years? x What percent of the students have cars that a new than 5 years old? x How likely is it that a student has a new car? Explain. x If a student wants to have a car that is newer than 70% of the cars at the high school, how old

should the car be?

8. The Beanstalk Club is limited to women and men who are very tall.  If women’s heights have a mean 

of 63.6 inches and a standard deviation of 2.5 inches. Find the cut off heights for the following chapters knowing that women’s heights are normally distributed.

x The Travis County chapter wants to take women who are taller than 1.75 standard deviations above the mean.

x The Austin chapter wants to take only women in the top 8%. x The Central Texas chapter wants to take women in the 88th percentile.

9. An educational testing service has designed a new test of mechanical aptitude. Scores on this test are

normally distributed with P� �400 and V = 60.

a) What score would you need to be in the top 15%? b) What score represents the 45th percentile? c) If 200 students at McCallum took the test, how many would you expect to score below 300?