Central Limit Theorem – 20 points

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Central Limit Theorem
For each question, either give the correct answer accurate to 4 decimal places (ex: 23.5%)
or if you feel that the question cannot be asked, explain why.
1. Credit card balances for young couples are roughly normally distributed and have a
mean of $750 and a standard deviation of $265.
a. What is the probability that a typical couple’s balance is more than $1,000?
b. What is the probability that the average balance of an SRS of 5 couples is more than
$1,000?
c. What is the probability that the average balance of an SRS of 100 couples is less than
$700?
2. The amount of snow that falls in Buffalo, NY over a winter is normally distributed
with a mean of 15 feet, 7 inches and a standard deviation of 6 feet, 2 inches.
a. What is the probability that Buffalo will have over 12 feet of snow next winter?
b. What is the probability that Buffalo’s next 4 winters will average over 10 feet of snow?
3. Tomato plants watered lightly for a month show an average growth of 27 centimeters
with a standard deviation of 8.3 centimeters. Assume that tomato plants grown according
to a normal distribution.
a. What is the probability that a plant will grow over 30 centimeters?
b. What is the probability that an SRS of 50 plants will have an average growth of over
30 centimeters?
c. Compare answer b to answer a. Explain why answer b is (smaller/greater) than answer
a.
d. What is the probability that a plant will grow between 25 and 30 centimeters?
e. What is the probability that 100 plants will have an average growth between 25 and 30
centimeters?
f. Compare answer e to answer d. Explain why answer e is (smaller/greater) than answer
d.
4. The average amount of money spent at lunch in the West Allegheny cafeteria is $3.00
with a standard deviation of 75 cents. Assume the distribution of money spent is normal.
a. What is the probability that a student spends more than $3.50 for lunch?
b. What is the probability that the average amount spent by an SRS of 10 students will be
greater than $4?
c. What is the probability that a student spends less than $2.75 for lunch?
d. Find the probability that the average amount spent by an SRS of 25 students is less
than $2.75.
5. The typical 6 ounce bag of potato chips is normally distributed by weight with a
standard deviation of .15 ounces.
a. What is the probability that a bag contains less than 5.9 ounces?
b. What is the probability that the average weight of an SRS of 12 bags will be less than
5.9 ounces?
c. What is the probability that a bag will have more than 6.2 ounces or less tan 5.8
ounces?
d. Find the probability that an SRS of 5 bags has more than 6.2 ounces or less than 5.8
ounces.
6. The average number of years that a particular washing machine lasts is 7.45 years with
a standard deviation of 2.71 years. Assume normality. The warranty for this machine is
two years.
a. What is the probability that a machine will last more than 8 years?
b. What is the probability that an SRS of 100 washing machines will last more than 8
years?
c. What is the probability that a machine will fail within the warranty period?
d. What is the probability that an SRS of 15 will all fail within the warranty period?
7. The average amount of time that people spend going through airport security for planes
taking off between 8 AM and 10 AM at a busy airport is 21 minutes with a standard
deviation of 4.2 minutes.
a. What is the probability that a person has to wait more than 25 minutes?
b. What is the probability that the average wait for the next 8 people in line have to wait
is more than 25 minutes?
c. What is the probability that an SRS of 75 people will wait an average of 20 minutes or
less?
5. An SAT review course claims that it can increase SAT scores with great success. It
reports that the average gain of a student is 50 points with a standard deviation of 21.2.
No other information is given.
a. If a student takes the course, what is the probability that her scores will increase 50
points or more?
b) If a student takes the course, what is the probability that her scores will increase 55
points or more?
c) If 10 students take the course, what is the probability that the average gain will be 55
points or more?
d) If 100 students, take the course, what is the probability that the average gain will be 55
points or more?
e) If 200 students, take the course, what is the probability that the average gain will be 55
points or more?
f) Which of these answers are you most confident of? Why?
8. At 8 AM on a typical weekday morning, the average number of people in a 7-11 store
is 24.5 with a standard deviation of 4.4.
a. What is the approximate distribution of the mean number of persons x in 500
randomly selected 7-11’s?
b. What is the probability that 500 selected stores will have more than 12,500 people in
them at 8 AM?
9. A survey of 1150 people found 972 believed the Steelers would win the Super Bowl.
a. What is the population proportion who believe the Steelers will win the Super Bowl/
b. What is the probability a simple random sample of 100 people will show that less than
79% believe the Steelers will win the Super Bowl?
c. Find an interval where 97% of the values should lie?
d. The survey says it is accurate within plus or minus 4% . Is this true?
10. The Pittsburgh Post Gazette reports almost 75% of all people like nuts or caramel in
their chocolate. A random sample of 200 people is taken and the number who like nuts
or caramel in their chocolate is recorded.
a. What is the approximate sampling distribution for the sample proportion p̂ ?
b. What are the mean and standard deviation for the sample?
c. What is the probability the sample proportion is greater than 80%?
d. What is the probability the sample proportion is less than 72%?
e. Within what limits would you expect 95% of values to lie?
f. Within what limits would you expect 99% of values to lie?
g. What value is greater than 92% of all values?
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