Exercises. 2. If X is normally distributed with a mean of 30 and a standard deviation

1. If , find

a) ; b) );

c) ; d)

2. If X is normally distributed with a mean of 30 and a standard deviation

of 5, find

a) ; b) ;

c) ; d)

3. If X has a normal distribution with and , find b such that

a) ;

b) ;

c)

4. Scores on a university entrance examination follow a normal distribution with a mean of 500 and a standard deviation of 100. Find the probability that a student will score

a) over 650;

b) less than 250;

c) between 325 and 675;

d) If the university admits students who score over 670, what proportion of the student pool would be eligible for admission?

e) What should be the limit if only the top 15% are to be eligible?

5. According to the children’s growth chart that doctors use as a reference, the heights of two-year-old boys are nearly normally distributed with a mean of 85 cm inches and a standard deviation of 5 cm. If a two year-old boy is selected at random, what is the probability that he will be between 75 cm and 92 cm tall?

6. The weights of apples served at a restaurant are normally distributed with

a mean of 125 grams and standard deviation of 8 grams. What is the

probability that the next person served will be given an apple that weights

less than 120 grams?

7. The National bank is reviewing its service charge and interest-paying policies on checking accounts. The bank has found that the average daily balance on personal checking accounts is $55000, with a standard deviation of $15000. In addition, the average daily balances have been found to be normally distributed.

a) What percentage of personal checking account customers carry average daily balances in excess of $80000?

b) What percentage of the bank’s customers carry daily balances below $20000?

c) What percentage of the bank’s customer carry average daily balances between $30000 and $70000?

d) The bank is considering paying interest to customers carrying average daily balances in excess of a certain amount. If the bank does not want to pay interest to more than 5% of its customers, what is the minimum average daily balance it should be willing to pay interest on?

8. The sales of high-bright toothpaste are believed to be approximately normally distributed, with a mean of 10 000 tubes per week and a standard deviation of 1500 tubes per week.

a) What is the probability that more than 12000 tubes will be sold in any given week?

b) In order to have a 0.95 probability that the company will have sufficient stock to cover the weekly demand, how many tubes should be produced?

9. The attendance at football games at a certain stadium is normally distributed, with a mean of 45000 and a standard deviation of 3000.

a) What percentage of the time should attendance be between 44000 and 48000?

b) What is the probability of exceeding 50000?

c) Eighty percent of the time the attendance should be at least how many?

10. The lifetime of a color television picture tube is normally distributed, with a mean of 7.8 years and a standard deviation of 2 years.

a) What is the probability that a picture tube will last more than 10 years?

b) If the firm guarantees the picture tube for 2 years, what percentage of the television sets sold will have to be replaced because of picture tube failure?

c) If the firm is willing to replace the picture tubes in a maximum of 1% of the television set sold, what guarantee period can be offered for the television picture tubes?

11. It is estimated that the scores on the university entrance test are distributed normally with mean of 80 and standard deviation of 5.

a) For a randomly chosen participant taking this test, what is the probability of a score more than 72?

b) For a randomly chosen participant taking this test, what is the probability of a score between 73 and 85?

c) What is the minimum score needed in order to be in the top 5% of all participants taking this test?

d) What is the minimum score is needed to enter to the university if only the best 70% of all participants will pass this test?

e) Two participants are chosen at random. What is the probability that at least one of them scores more than 85?

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