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More Central Limit Theorem Practice
1. In a certain school district the distribution of the heights of eighth-graders who play the tuba
is approximately normal, with a mean of 146.1 cm and a standard deviation of 8.4 cm. Suppose
you take a sample of 5 eighth-graders.
a) What is the probability that the average height for this sample is less than 150 cm?
b) Among all samples of 5 eighth-graders, 63% of these samples have mean heights greater than
what?
2. In a study for an acne medication, dermatologists count the number of blemishes on patients’
faces. They found the mean number of blemishes was 9.2 with a standard deviation of 3.1.
a) Suppose you take a sample of 10 teenagers. What is the probability that the average number
of blemishes is more than 10?
b) Suppose you take a sample of 40 teenagers. What is the probability that the average number
of blemishes is more than 10?
3. The weights of the eggs produced by a certain breed of hen are symmetrically distributed –
although only roughly Normal – with mean 65 grams and standard deviation 5 grams.
a) What is the probability that one egg selected at random from a hen house will weigh more
than 68 grams?
b) How accurate do you think your calculation from part (a) is and why?
c) Consider a carton of 18 eggs to be a simple random sample of hen’s eggs. If you were to take a
large number of repeated samples of size 18, what would the mean and standard deviation be of
these sample means?
d) What is the probability that the average weight in a carton of 18 eggs is more than 68 grams?
e) What is the probability that the average weight in a package of 36 eggs is more than 68
grams?
f) Of all the calculations you did for this problem – which are you most confident in and why?
4. In a study done on the life expectancy of 500 people in a certain geographic region, the mean
age at death was 72 years old and the standard deviation was 5.3 years.
a) What is the probability that an individual selected at random will be less than 70 years old
when they die?
b) If a sample of 50 people from this region is selected, find the probability that the mean life
expectancy will be less than 70 years.
c) For samples of size 50, what life expectancy will 80% of sample means be less than ?
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