Download Chapter 8 Sampling Distributions – Sample

Name: _______________________________
AP Statistics
Chapter 8 Review Worksheet
1) Taxi fares are normally distributed with mean fare $22.27 and standard deviation $2.20.
a) Which should have the greater probability of falling between $21 and $24 – the mean of 10 random taxi
fares or the amount of a single random taxi fare? Why?
b) Which should have a greater probability of being greater than $24 – the mean of 10 random taxi fares or the
amount of a single taxi fare? Why?
2) Suppose a sample of n = 50 items is drawn from a population of manufactured products and the weight, x, of
each item is recorded. Prior experience has shown that the weight has a mean of 6 ounces and standard
deviation of 2.5 ounces.
a) What is the shape of the sampling distribution of x ?
b) What is the mean and standard deviation of the sampling distribution?
c) What is the probability that the manufacturer’s sample has a mean weight of less than 5 ounces? If you did
get a sample mean of 5 or less, does this indicate that the manufacturing process may be faulty? Explain.
d) How would the sampling distribution of x change if the sample size, n, were increased from 50 to 100?
3) A soft-drink bottle vendor claims that its production process yields bottles with a mean internal strength of
157 psi (pounds per square inch) and a standard deviation of 3 psi and is normally distributed. As part of its
vendor surveillance, a bottler strikes an agreement with the vendor that permits the bottler to sample from the
vendor’s production to verify the vendor’s claim.
a) Suppose the bottler randomly selects a single bottle to sample. What is the mean and standard deviation?
b) What is the probability that the psi of the single bottle is 1.3 psi or more below the process mean?
c) Suppose the bottler randomly selected 40 bottles from the last 10,000 produced. Describe the sampling
distribution of the sample means?
d) What is the probability that the sample mean of the 40 bottles is 1.3 psi or more below the process mean?
e) In order to reduce the standard deviation 50% (half), how large would the sample size need to be?
4) In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons.
Furthermore, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and
staff on campus is 150 pounds, that the standard deviation is 27 pounds, and that the distribution of weights of
individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken:
a) What is the expected value of the sample mean of their weights?
b) What is the standard deviation of the sampling distribution of the sample mean weight?
c) What average weights for a sample of 16 people will result in the total weight exceeding 2500 pounds?
d) What is the probability that a random sample of 16 persons on the elevator will exceed the weight limit?
Name: _____________________________________
AP Statistics
Normal Distributions Review WS
EX) The army reports that the distribution of head circumference among soldiers is approximately
normal with mean 22.8 inches and standard deviation of 1.1 inches.
a) What is the probability that a randomly selected soldier’s head will have a circumference that is
greater than 23.5 inches?
b) What is the probability that a randomly selected soldier’s head will have a circumference that is between 20
than 23 inches?
c) How many inches in circumference would a soldier’s head be if it were at the 40th percentile?
d) Helmets are mass-produced for all except the smallest 5% and the largest 5%. Soldiers in the smallest and
largest 5% get custom-made helmets. What head sizes get custom-made helmets?
1) Here are the prices (in cents per pound) of bananas from 15 markets surveyed by the U. S.
Department of Agriculture. Are banana prices normally distributed? Justify your answer.
51
52
45
48
53
52
50
49
52
48
43
46
45
42
50
2) A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose the capacity of gas tanks are
normally distributed with mean of 15 gallons and standard deviation of 0.1 gallon.
a) What is the probability that a randomly selected tank will hold at most 14.8 gallons?
b) What is the probability that a randomly selected tank will hold between 14.7 and 15.1 gallons?
c) How many gallons would a tank hold if it were at the 3 rd quartile of this distribution?
d) If two gas tanks are independently selected, what is the probability that both tanks hold at most 15 gallons?
3) Suppose that for the population of students at a particular university, the time required to
complete a standardized exam is normally distributed with mean 45 minutes and standard deviation 5
minutes.
a) If 50 minutes is allowed for the exam, what proportion of students at this university would be unable to
finish in the allotted time?
b) What proportion of students at this university would finish the exam in more than 38 minutes, but less than
48 minutes?
c) How much time should be allowed for the exam if we wanted 90% of the students taking the test to be able
to finish in the allotted time?
d) How much time is required for the fastest 25% of all students to complete the exam?
Chapter 8 Sampling Distributions – Sample Means
Definitions:
Parameter –
Statistic –
Example: Identify the boldface values as parameter or statistic. (YMM p. 457)
A carload lot of ball bearings has mean diameter 2.5003 cm. This is within the specifications for acceptance of
the lot by the purchaser. By chance, an inspector chooses 100 bearings from the lot that have mean diameter
2.5009 cm. Because this is outside the specified limits, the lot is mistakenly rejected.
Why do we take samples instead of taking a census?
The _____________________ of a statistic is the
distribution of values taken by the statistic in all possible
samples of the same size from the same population.
A statistic used to estimate a parameter is _______________
if the mean of its sampling distribution is equal to the true
value of the parameter being estimated.
If n is large or the population distribution is normal, then
z 
x  x
x

has approximately a standard normal distribution.
Consider the population – the length of fish (in inches) in my pond - consisting of the values : 2, 7, 10, 11, 14
  _______________
  _______________
Let’s take samples of size 2 (n = 2) from this population:
Pairs
x
 x  _______________
 x  _______________
Repeat this procedure with sample size n = 3
Pairs
x
 x  _______________
 x  _______________
General Properties of the Sampling Distribution of x
Rule 1:
Rule 2:
Rule 3:

The standard deviation of the sampling distribution of the mean is referred to as the standard error of
the mean. If random samples of size n, where n is more than 5% (10%) of the population size (N), are
selected from a population whose standard deviation is , then
Rule 4:
Ex 1) The army reports that the distribution of head circumference among soldiers is approximately normal with
mean 22.8 inches and standard deviation of 1.1 inches.
a) What is the probability that a randomly selected soldier’s head will have a circumference that is greater than
23.5 inches?
b) What is the probability that a random sample of five soldiers will have an average head circumference
that is greater than 23.5 inches?
Ex 2) Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the
length of a single trout taken at random from the pond. This group of biologists has determined that the length
has a normal distribution with mean of 10.2 inches and standard deviation of 1.4 inches.
a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?
b) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long?
c) What sample mean would be at the 95th percentile? (Assume n = 5)
Ex 3) A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume
of soda in a randomly selected can. Suppose that x is normally distributed with
s = .16 oz. Sixteen cans
are to selected with a mean of 12.1 oz.
a) What is the probability that the average of 16 cans will exceed 12.1 oz?
b) Do you think the bottler’s claim is correct?
Ex 4) A hot dog manufacturer asserts that one of its brands of hot dogs has a average fat content of 18 grams
per hot dog with standard deviation of 1 gram. Consumers of this brand would probably not be disturbed if the
mean was less than 18 grams, but would be unhappy if it exceeded 18 grams. An independent testing
organization is asked to analyze a random sample of 36 hot dogs. Suppose the resulting sample mean is 18.4
grams. Does this result indicate that the manufacturer’s claim is incorrect?
What if the sample mean was 18.2 grams, would you think the claim was incorrect?
Homework:
For questions 1 & 2, identify the boldface values as parameter or statistic
1) A researcher carries out a randomized comparative experiment with young rats to investigate the effects of
a toxic compound in food. She feeds the control group a normal diet. The experimental group receives a diet
with 2500 parts per million of the toxic material. After 8 weeks, the mean weight gain is 335 grams for the
control group and 289 grams for the experimental group.
2) A telemarketing firm in Los Angeles uses a device that dials residential telephone numbers in that city at
random. Of the first 100 numbers dialed, 48% are unlisted. This is not surprising because 52% of all Los
Angeles residential phones are unlisted.
3) State tax officials claim that the average amount of money claimed by all the taxpayers within the state for
charitable deductions during 1994 was $964 with a standard deviation of $102. Many samples of size 64 are
taken. Find the mean of these samples and the standard error of the mean.
4) The scores of six students on an exam are as follows.
Student
Score
1
82
2
62
3
80
4
57
5
72
6
73
Assume that this is your population, create the sampling distribution for the means of sample size n = 4. Find
the mean and standard deviation of the sampling distribution. (Hint: there are 15 possible samples of size 4.
5. The weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams (g)
and standard deviation 5 g.
(a) Calculate the probability that a randomly selected egg weighs between 62.5 g and 68.75 g. Show
your work.
Think of cartons of such eggs as SRSs of size 12 from the population of all eggs.
(b) Calculate the probability that the mean weight of a carton falls between 62.5 g and 68.75 g. Show
your work.
(c) Did you need to know that the population distribution of egg weights was Normal in order to
complete parts (a) or (b)? Justify your answer.