Download Sampling Distribution Exercises

Sampling Distribution Exercises
1.
Suppose a random sample of n measurements is selected from a population with mean
100 and standard deviation 10. For each of the following values of n, give the mean and
standard deviation of the sample distribution of the sample mean x .
a) n = 4 b) n = 25 c) n = 100
d) n = 50 e) n = 500 f) n = 1000
2.
A random sample of size n = 64 observations is drawn from a population with a mean
equal to 20 and standard deviation equal to 16
a) Give the mean and standard deviation of the sampling distribution of x .
b) Describe the shape of the sampling distribution of x . Does you answer depend on
the sample size?
c) Calculate the standard normal z-score corresponding to a value of x = 15.5
d) Find the probability that x is less than 16.
e) Find the probability that x falls between 16 and 22
3.
At a large factory, the mean wage is $42,500 and the standard deviation is $2000. What is
the probability that the mean wage of 75 randomly selected workers will exceed $43,000?
4.
At a city high school, past records indicate that the MSAT scores for students have a mean
of 510 and standard deviation of 90. One hundred students in the high school are to take
the test. What is the probability that their mean score will be
(a) More than 530?
(b) Less than 500?
(c) Between 495 and 515?
5.
An appliance manufacturer claims that the mean life of its product is 1200 hours.
Assume that the standard deviation is 120 hours. A consumer agency decided to
randomly select 35 items and will reject the claim if x < 1160 hours. If the manufacturer's
claim is true, what is the probability that the claim will be rejected?
6.
A grocery store produce manager is told by a wholesaler that the apples in a large
shipment have a mean weight of 6 ounces and a standard deviation of 1 ounce. The
manager is going to randomly select 100 apples.
(a) Assuming the wholesaler's claim is true, find the probability that the mean weight of
the sample is more than 5.9 ounces.
(b) The manager decides to return the shipment if the mean weight of the sample is less
than 5.75 ounces. Assuming the wholesaler's claim is true, find the probability that
the shipment will be returned.
7.
The heights (x) of players in a division of high school football teams are approximately
normal with mean 71 inches and standard deviation 2.5 inches. Consider the distribution
of sample means with sample size n=100.
(a) What percentage of sample means are larger than 70.5?
(b) What percentage of heights are more than 70.5?
8. Measurements in the lab: Juan makes a measurement in a chemistry laboratory and records
the result in his lab report. The standard deviation of students’ lab measurements is   10
milligrams Juan repeats the measurement 3 times and records the mean x of his
measurements.
(a) What is the standard deviation of Juan’s mean result?
(b) How many times must Juan repeat the measurement to reduce the standard deviation of
x to 5?
9.
Explain what happens to the mean and standard deviation of x when the sample size
is increased or decreased.
10. The scores of 12-th grade students on the National Assessment of Educational
Progress year 2000 mathematics test have a distribution that is approximately
Normal with mean 300 and standard deviation 35.
a) Choose one 12th grader at random. What is the probability that his or her score is
higher than 320
b) Now choose a random sample of four 12th graders. What is the probability that their
mean score is higher than 320?
11. An insurance company sees that in the entire population of homeowners, the mean
loss from fire is $250 and the standard deviation of the loss is $1000. The
distribution of losses is strongly right-skewed: many policies have $0 loss, but a
few have large losses. If the company sells 10,000 policies, what is the
approximate probability that the average loss will be greater than $275?
12. The weight of eggs produced by a certain breed of hen is normally distributed with
mean 65 grams and standard deviation 5 grams. Think of cartons of eggs as simple
random samples of size 12 from the population of all eggs. What is the probability
that the weight of the carton falls between 750 g and 825 g?
13. A grocery store produce manager is told by a wholesaler that the apples in a large
shipment have a mean weight of 6 ounces and standard deviation 1 ounce. The
manager is going to randomly select 100 apples.
Suppose the retailer is willing to risk a 1% chance of returning the shipment if the
wholesaler’s claim is true. Let W be the mean weight of the sample below which the
shipment will be returned. Find the value of W.
14. A dairy claims that the mean amount of milk in its milk containers is 128 ounces.
Let x be the number of ounces of milk per container, and assume that x is normally
distributed with standard deviation of 1 ounce. A random sample of 25 containers
gave a sample mean of 127.4 ounces.
Find P ( x  127.4) . Do you think there that is evidence that the true mean is less
than 128 ounces?
15. A company has been producing a 6-watt light bulb with a mean life of 750 hours and a
standard deviation of 30 hours. An engineer developed a new process for producing the
bulb. It was believed that bulbs produced by the new process would show the same
standard deviation but not possible a longer mean lifetime. Thirty-six bulbs produced by the
new process were tested and showed a sample mean lifetime of 765 hours.
(a) Assuming that the population mean lifetime  of bulbs produced by the new
process is still 750 hours, find the probability of getting a value of the sample
mean as large or larger than 765.
(b) What conclusions might be drawn from the answer to part (a)?