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Math 116 – Chapter 11 - Central Limit Theorem – problems from another book
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When finding probabilities, practice both ways, showing steps, and with the calculator feature
1) The mean annual income for adult women in a certain city is $28,520 and the standard deviation is
$5600. Considering that the distribution of incomes is right skewed, determine the distribution of sample
means for samples of size 95.
a) x-bar is approximately normal with mean = $28520 and standard deviation $5600
b) x-bar is right skewed with mean = $28520 and standard deviation $5600
c) x-bar is approximately normal with mean = $28520 and standard deviation $575
d) x-bar is right skewed with mean = $28520 and standard deviation $575
2) Suppose x has a distribution with mu = 80 and sigma = 8
a) If random samples of size n = 16 are selected, describe the shape, mean and standard deviation of the
distribution of sample means for samples of this size.
b) If random samples of size n = 36 are selected, describe the shape, mean and standard deviation of the
distribution of sample means for samples of this size.
3) Suppose x has a normal distribution with mu = 75 and sigma = 10
a) If random samples of size n = 16 are selected, describe the shape, mean and standard deviation of the
distribution of sample means for samples of this size.
b) If random samples of size n = 36 are selected, describe the shape, mean and standard deviation of the
distribution of sample means for samples of this size.
4) Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long
train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of
coal loaded into each car are normally distributed, with mean mu = 75 and standard deviation sigma =
0.8 tons.
a) What is the probability that one car chosen at random will have less than 74.5 tons of coal?
b) What is the probability that 20 cars chosen at random will have a mean load weight x-bar of less than
74.5 tons of coal?
c) Suppose the weight of coal in one car was less than 74.5 tons. Would that fact make you suspect that
the loader had slipped out of adjustment? Why?
d) Suppose that the weight of coal in 20 cars selected at random had an average x-bar lf less than 74.5.
Would that fact make you suspect that the loader had slipped out of adjustment? Why?
5) Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of
blood) after a 12-hour fast. Assume that for people under 50 years old, x has a distribution that is
approximately normal, with mean mu = 85 and estimated standard deviation sigma = 25. A result of x <
40 is an indication of severe excess insulin, and medication is usually prescribed.
a) What is the probability that, on a single test, x < 40?
b) Suppose a doctor uses the average x-bar from two tests taken about a week apart. What can we say
about the probability distribution of x-bar? What is the probability that x-bar < 40?
c) Repeat part (b) for n = 3 tests taken a week apart.
d) Repeat part (b) for n = 5 tests taken a week apart.
e) Compare your answers to parts a, b, c, and d. How did the probabilities change as n increased? What
must this imply if you were a doctor? If a patient had a test result of x-bar < 40 based on 5 tests, explain
what must this imply?
6) Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood.
Assume that x has a distribution that is approximately normal, with mean mu = 7500 and estimated
standard deviation of sigma = 1750. A test result of x < 3500 is an indication of leucopenia. This
indicates bone marrow depression that may be the result of a viral infection.
a) What is the probability that, on a single test, x is less than 3500?
b) Suppose a doctor uses the average x-bar for two tests taken about a week apart. What can we say
about the probability distribution of x-bar? What is the probability of x-bar < 3500?
c) Repeat part (b) for n = 3 tests taken a week apart.
d) Compare your answers to parts a, b, and c. How did the probabilities change as n increased? If a
person had x-bar < 3500 based on three tests, what conclusion would you draw as a doctor or a nurse?
ANSWERS
1) c
2) a) For the x-bar distribution when n = 16: We can’t specify the shape, the mean will be 80 and the standard
error will be 2
b) For the x-bar distribution when n = 36: The shape will be approximately normal, the mean will be 80 and the
standard error will be 1.333
3) a) For the x-bar distribution when n = 16: The shape will be normal, the mean will be 75 and the standard
error will be 2.5
b) For the x-bar distribution when n = 36: The shape will be normal, the mean will be 75 and the standard error
will be 1.667
4) a) .2643, b).0026, c) No, we expect about 26% of the cars to have a weight less than 74.5 tons; d) Yes, we
expect only 0.26% of means of 20 cars to be less than 74.5 tons. The low probability says that either a very rare
event has happened or the loader has slipped out of adjustment. By chance, we expect 26 x-bars out of 10,000 to
be lower than 74.5
5) a) 0.0359, b) 0.0054, c) 0.0009; d) 0.00003; e) The probabilities decreased as n increased. It would be an
extremely rare event for a person to have and x-bar < 40 just by chance. It’s more likely that the person has a
case of excess insulin.
6) a) 0.0111; b) 0.0006; c) 0.00004; d) the probabilities decreased as n increased. It would be an extremely rare
event for the mean of two or three tests to be less than 3500. As a nurse I would think that this patient has
leucopenia.