8.1 Statistics and Sample Variability
Statistic
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Some statistics we will use include:
𝑥̅ :
s:
𝑝̂ :
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The campus of Wolf City College has a fish pond. Suppose there are 20 fish in the pond. The
lengths of the fish (in inches) are given below:
4.5
5.4
10.3
7.9
8.5
6.6
11.7
8.9
2.2
9.8
6.3
4.3
9.6
8.7
13.3
4.6
10.7
13.4
7.7
5.6
Suppose we caught three fish with lengths 6.3”, 2.2” and 13.3”
2nd sample: 8.5”, 4.6”, 5.6”
3rd sample: 10.3”, 8.9”, 13.4”
Sampling Distributions of 𝑥̅
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Suppose there are only 5 fish in the pond. The lengths of the fish (in inches) are:
6.6
11.7
8.9
2.2
9.8
What is the mean and standard deviation of this population?
Let’s find all the samples of size 2
Pairs
6.6
11.7
6.6
8.9
6.6
2.2
6.6
9.8
11.7
8.9
11.7
2.2
11.7
9.8
8.9
2.2
8.9
9.8
2.2
9.8
11.7
8.9
9.8
11.7
2.2
9.8
8.9
2.2
9.8
𝑥̅
What is the mean and standard deviation of these sample means?
Let’s find all the sample sizes of 3
Triples
6.6
11.7
8.9
6.6
11.7
2.2
6.6
11.7
9.8
6.6
8.9
9.8
6.6
8.9
9.8
6.6
2.2
9.8
11.7
8.9
2.2
𝑥̅
What is the mean and standard deviation of these sample means?
Mean and Standard Deviation of Sampling Distributions and Population
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General Properties of Sample Distributions of 𝑥̅
Rule 1:
Rule 2:
Rule 3:
Rule 4:
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 𝜎 = .16 oz. Sixteen cans are randomly selected, and the soda volume is determined for
each one. Let x = the resulting sample mean soda.
If the bottler’s claim is correct, then the sampling distribution of x is normally distributed
with:
𝜇𝑥̅ =
𝜎𝑥̅ =
What is the probability that the sample mean soda volume is between 11.96 ounces and
12.08 ounces?
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
suggest that the manufacturer’s claim is incorrect?
1.
For each of the following statements, identify the number that appears in boldface
type as the value of either a population characteristic or a statistic:
a. A department store reports that 84% of all customers who use the store’s credit plan pay
their bills on time.
b. A sample of 100 students at a large university had a mean age of 24.1 years.
c. The Department of Motor Vehicles reports that 22% of all vehicles registered in a
particular state are imports.
d. A hospital reports that based on the 10 most recent cases, the mean length of stay for
surgical patients is 6.4 days.
2.
Suppose we artificially categorize populations as approximately normal or not
approximately normal, and samples as large or small. This categorization results in 4
categories:
a) Small samples from an approximately normal population
b) Large samples from an approximately normal population
c) Small samples from a population that is not approximately normal
d) Large samples from a population that is not approximately normal
What, if anything, can be said about the shape of the sampling distribution of for each of
these 4 situations?
(a)
(b)
(c)
(d)
3.
The Get-A-Grip tire company claims that the mean lifetime of tires sold on new cars
is 23,000 miles and the standard deviation is 2500 miles.
a) If the claim by Get-A-Grip is true, what is the mean of the sampling distribution of x
for samples of size n = 4 ?
b) If the claim by Get-A-Grip is true, what is the standard deviation of the sampling
distribution of x for samples of size n = 4 ?
c) If the distribution of tire life is approximately normal, what is the probability that the
mean of a random sample of n = 4 tire lifetimes will be less than 20,000 miles?
4.
For the given pairs ( x , s ) and ( µ , σ ) which phrase best describes the difference
between the two:
(A) These two pairs are essentially the same.
(B) These two pairs use the same formulas but n changes.
(C) The first pair represents mean and standard deviation for a sample, while the
second represent mean and standard deviation for a population.
(D) The first pair represents mean and standard deviation for a population, while the
second represents mean and standard deviation for a sample.
(E) The first pair is represents mean and standard deviation, but the second pair
represents standard deviation and mean.
Sample Distribution of 𝑝̂
p:
𝑝̂ :
Suppose we have a population of six students: Alice, Ben, Charles, Denise, Edward &
Frank. We are interested in the proportion of females. What is the proportion of females?
Let’s select samples of two from this population
Alice & Ben
Ben & Frank
Alice & Charles
Charles & Denise
Alice & Denise
Charles & Edward
Alice & Edward
Charles & Frank
Alice & Frank
Denise & Edward
Ben & Charles
Denise & Frank
Ben & Denise
Edward & Frank
Ben & Edward
Find the mean and standard deviation of these sample proportions.
General Properties for Sampling Distributions of 𝑝̂
Rule 1:
Rule 2:
Rule 3:
Blood Transfusions
Let p = proportion of patients who contract hepatitis after a blood transfusion
p = .07
Suppose a new blood screening procedure is believed to reduce the incident rate of hepatitis. Blood
screened using this procedure is given to n = 200 blood recipients. Only 6 of the 200 patients
contract hepatitis. Does this result indicate that the true proportion of patients who contract
hepatitis when the new screening is used is less than 7%?
Is the sampling distribution approximately normal?
𝜇𝑝̂ =
𝜎𝑝̂ =
Does this result indicate that the true proportion of patients who contract hepatitis when the new
screening is used is less than 7%?
5.
Consider sampling from a population whose proportion of successes is pˆ = 0.5 . As the
sample size, n, increases, some characteristics of the sampling distribution of change. Which of the
following characteristics will change as n increases, and what is the nature of the change?
a) The mean of the sampling distribution of p̂
b) The standard deviation of the sampling distribution of p̂
c) The shape of the sampling distribution of p̂
6.
A local owner of apartment buildings is considering raising the rent because of increased
costs. He is concerned that apartments may be easy to come by, and will only increase the rent if
the current vacancy rate is below 20%. One method for estimating vacancy rate is to take a random
sample of similar apartments, and calculate the proportion of apartments that are empty. Suppose
the owner has decided to take a sample to estimate the vacancy rate. He is considering a sample
size of n = 40. Suppose further that the actual vacancy rate is 0.30.
a) Show that this sample size is large enough for the sampling distribution of p̂ to be
approximately normal.
b) What is the mean of the sampling distribution of p̂ ?
c) What is the standard deviation of the sampling distribution of p̂ ?
d) What is the approximate probability that a sample proportion, p̂ , would be less than 0.20 even
though the population proportion is 0.30?
7. A random sample of the costs of repair jobs at a large muffler repair shop produces a mean of
$127.95 and a standard deviation of $24.03. If the size of this sample is 40, which of the
following is an approximate 90 percent confidence interval for the average cost of a repair at this
repair shop?
A.
B.
C.
D.
E.
$127.95±4.87
$127.95±6.25
$127.95±7.45
$127.95±30.81
$127.95±39.53
8. The article “Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to
Toxic Waste Sites” (Chance [1992] : 40 – 45) reported that in a large study carried out in the
state of New York, approximately 30% of the study subjects lived within 1 mile of a hazardous
waste site. Let π denote the proportion of all New York residents who live within 1 mile of such
a site and suppose that π – 0.3.
a. Would p based on a random sample of only 10 residents have approximately a normal
distribution? Explain why or why not.
b. What are the mean value and standard deviation of p based on a random sample of size 400?
c. When n = 400, what is P(0.25 ≤ p ≤ 0.35)?
d. Is the probability calculated in Part (c) larger or smaller than would be the case if n = 500?
Answer without actually calculating this probability.
9. Suppose that a customer requires at least 110 ropes that can withstand up to 400 pounds of
force. Suppose further that 88% of ropes that a certain rope company produces can withstand up
to 400 pounds of force. How many ropes does the customer need to buy to be 98% confident that
they will have at least 110 ropes that can withstand up to 400 pounds of force?
(A) 134
(B) 133
(C) 125
(D) 117
(E) 116