# Statistics

Problems on Normal Distribution

Directions: Draw a clearly labeled diagram and then solve each problem.

1. The gestation period (length of pregnancy) for male babies born in New York is normally distributed with a mean of 39.4 weeks and a standard deviation of 2.43 weeks.

(a) What percent of mothers of male babies are pregnant for less than 35 weeks?
(b) What percent of mothers of male babies are pregnant for between 35 and 40 weeks?
(c) What percent of mothers of male babies are pregnant for more than 40 weeks?
(d) Suppose a sailor had shore leave 46 weeks ago and his wife is delivering a baby today. What should the sailor say to his wife when he gets home?

2. The daily demand for gas at Good’s Gas station is normally distributed with a mean of 1812 gallons and a standard deviation of 254 gallons.

(a) What is the probability that the demand for gas will exceed 2000 gallons on any day?
(b) What is the probability that the demand for gas in a day will be between 1500 and 2000 gallons?
(c) What is the probability that the demand for gas will exceed 1500 gallons on any day?

3. Suppose that the life of an Evercharged battery is normally distributed with a mean life of 12 hours and a standard deviation of 0.72 hours. What percent of Evercharged batteries will last:
a) less than 11.5 hours
b) more than 13 hours
c) between 11.5 and 13 hours?

4. Suppose that a school district wants to start a program for gifted students. The participants in the program are to be chosen on the basis of IQ scores (normal with  = 100,  = 15). If the school district wants only the top 2% of students to participate in the program, what should be the IQ score that students must exceed to be accepted?

5. In problem #2, how much gasoline should the station have on hand at the beginning of the day so that the probability of running out of gas that day is only 1%?

6. 10% of Americans are left handed.

(a) If 6 people are selected at random, what is the probability that more than 3 of them are left-handed?
(b) Suppose a group of 600 mathematicians get together for a conference. What is the probability that more than 80 of them are left-handed? (Use the normal approximation to the binomial)

7. Airlines sell more tickets for a flight than the number of available seats (overbooking). They do this because they know from past experience that only 90% of ticketed passengers actually show up for the flight.
(a) A plane has 6 seats. If the airline sells 8 tickets for a flight, what is the probability that the flight will be overbooked (the number of passengers who show up is greater than the number of available seats)?

(b) A plane has 325 seats. If the airline sells 350 tickets for a flight, what is the probability that the flight will be overbooked (the number of passengers who show up is greater than the number of available seats)?

8. The mean pulse rate for adults is 72 beats per minute (www.healthepic.com) and let’s suppose that the standard deviation is 10. Find:
a. The probability that a randomly chosen adult has a pulse rate over 80 assuming that the rates are normally distributed.
b. The probability that a random sample of 19 adults will have a mean score over 80.