Normal Distributions Homework Solutions
1. Most IQ tests have a mean of 100 and a standard deviation of 15. What percent of the population has an IQ greater than 130? Between 100 and 130? Greater than 145?
Answers: p (130 < x ) = : 02275 = 2 : 275% p (100 < x < 130) = 47 : 8% p (145 < x ) = : 00135 = 0 : 135%
2. Andr Lundberg has devised a di®erent IQ test.
(You can try it yourself at http://user.tninet.se/ ywq275g/index2.htm)
The mean is 35 and the standard deviation is 9.8. What score on the test in problem 1 corresponds to a score of 53 on Andr's test?
Answer: p (53 < x ) = : 03312 with m = 35 and s = 9 : 8
Now you need to solve p ( a < x ) = : 033 for a with m = 100 and s = 15 : You should get a = 127 : 5 approximately.
3. The physics department has three placement tests. On each test the mean is 100. On Test #1 the standard deviation is 10. On Test #2 the standard deviation is 8, and on Test #3 the standard deviation is 15.
Rank the following four results:
Student Test Score
Sandy 1 112
Chris
Kelly
Lou
2
3
3
110
112
125

Answer:
Lou had the highest score, then Chris, Sandy, and Kelly in that order. You could use a calculator and compute p ( a < x ) with a = 112 ; m = 100 ; s = 10 for Sandy, etc. However, you don't need to use a calculator. Just notice that Sandy is 1.2 deviations above the mean, Chris is 1.25 deviations above the mean, Kelly is .8 deviations above the mean, and Lou is 1 and 2/3 deviations above the mean.
4. According to the AAA, last year the average American spent $6,500 in driving expenses. Let's assume that these numbers follow a normal distribution with standard deviation $1500. Use the chart to answer questions 1 - 5.
1. What percentage spend less than $6500?
2. What percentage spend between $6500 and $8000?
3. What percentage spend more than $8000?
4. What percentage spend between $3500 and $8000?
5. What percentage spend less than $2000 ?
7. What percentage spend more than $7000?
8. What percentage spend between $5000 and $7000.
9. The top 10% spend at least how much money?
5. A population of an exclusive yuppie village is normally distributed with a mean age of 28 and a standard deviation of 4.6. Find the probability that a person chosen at random is between 25 and
35. Suppose the population of the village is 500,000. How many people are between 25 and 35?

Answer:
With m = 28 and s = 4 : 6 ; p (25 < x < 35) ¼ : 6788 : : :
500 ; 000 £ : 6788 ¢ ¢ ¢ = 339 ; 409
6. At Dalinde Hospital the birthweight of all newborn babies is normally distributed with a mean of
7.5 pounds and with a standard deviation of 0.5 pounds. What percent of babies weighed between
7 and 8 pounds?
Answer:
Use your calculator or Excel to ¯nd that p (7 < x < 8) ¼ : 6827 so as a percent, about 68.27 %.
7. The results of a state exam were normally distributed with a mean of 75 and a standard deviation of 8. Those in the top 5% received a special certi¯cate. Dale scored an 89. Does Dale get a special certi¯cate?
Answer:
With m=70 and s=8, p (89 < x ) ¼ : 04 ; so Dale was in the upper 4%. So Dale is entitled to a certi¯cate.
8. Given a normal distribution with a mean of 70 and standard deviation of 10, ¯nd the value of b so that p ( x < b ) = : 9
Answer: If you use a TI-83, you would use the function invNorm and enter invNorm(.9,70,10) and the number 82.81551567 should appear on the screen. You can use the function norminv in Excel.
To do that, you would enter =norminv(.9,70,10) in a cell to get the value b = 82 : 81550794 ¼ 83

9. Professor Hardy gave an exam where the scores were normally distributed. The average was 65 and the standard deviation was 16. He wants approximately 25 % to get a grade of E (a fail.) Find the lowest possible passing grade.
Answer: We need to ¯nd b such that p ( x < b ) = : 25 If you use a TI-83, you can use the function invNorm. You enter invnorm(.25,65,16). The answer is approximately 54.2 so Professor Hardy will assign a grade of E to a score of 54 or lower.