Download Online 14 - Section 7.2

Online 14 - Section 7.2-Doug Ensley
Student: _____________________
Date: _____________________
Instructor: Doug Ensley
Course: MAT117 01 Applied Statistics Ensley
Assignment: Online 14 - Section 7.2
1. According to a recent survey, the population distribution of number of years of education for self-employed individuals in a
certain region has a mean of 13.4 and a standard deviation of 4.9.
a. Identify the random variable X whose distribution is described here.
b. Find the mean and the standard deviation of the sampling distribution of x for a random sample of size 49. Interpret them.
c. Repeat (b) for n = 196. Describe the effect of increasing n.
a. Choose the correct description of the random variable.
A. The number of years of education
B. The number of people surveyed
C. The ages of the individuals
D. The number of self-employed individuals
b. The mean of the sampling distribution of size 49 is
. (Type an integer or a decimal.)
Choose the correct description of the mean of the sampling distribution.
A. The maximum mean for all samples of size 49
B. The mean of all samples of size 49
C. The expected value for the mean of a sample of size 49
D. The minimum mean for all samples of size 49
The standard deviation is
. (Type an integer or a decimal.)
Choose the correct description of the standard deviation.
A. The variablility of the mean for samples of size 49
B. The standard deviation of all samples of size 49
C. The maximum deviation of the mean for a sample of size 49
D. The minimum deviation of the mean for a sample of size 49
c. The mean of the sampling distribution of size 196 is
The standard deviation is
. (Type an integer or a decimal.)
The mean of the sampling distribution (1)
The standard deviation of the sampling distribution (2)
(1)
increases
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(2)
stays the same
decreases
decreases
stays the same
increases
ID: 7.2.14
. (Type an integer or a decimal.)
as n increases.
as n increases.
Online 14 - Section 7.2-Doug Ensley
2. For the population of farm workers in a certain country, suppose that weekly income has a distribution that is skewed to the
right with a mean of μ = $400 and a standard deviation of σ = $153. A researcher, unaware of these values, plans to
randomly sample 81 farm workers and use the sample mean weekly income x to estimate μ. Complete parts a through c
below.
Click here to view page 1 of an excerpt from a standard normal cumulative probability table.1
Click here to view page 2 of an excerpt from a standard normal cumulative probability table.2
a. What is the standard deviation of x?
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
b. Explain why it is almost certain that the sample mean will fall within $51 of $400. Choose the correct answer below.
A. The empirical rule says that if a distribution of data is bell shaped, then all or nearly all
observations fall within 2 standard deviations of the mean. The mean of a sample of such
observations would also fall within 2 standard deviations of the overall mean.
B. The empirical rule says that if a distribution of data is bell shaped, then all or nearly all
observations fall within 3 standard deviations of the mean. The mean of a sample of such
observations would also fall within 3 standard deviations of the overall mean.
C. The empirical rule says that if a distribution of data is bell shaped, then all or nearly all
observations fall within 51 units of the mean. The mean of a sample of such observations would
also fall within 51 units of the overall mean.
c. The sampling distribution of x provides the probability that x falls within a certain distance of μ, regardless of the value of
μ. Calculate the probability that x falls within $23 of μ for all such workers.
(Round to four decimal places as needed.)
1: Standard Normal Cumulative Probability Table Excerpt (Page 1)
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Online 14 - Section 7.2-Doug Ensley
2: Standard Normal Cumulative Probability Table Excerpt (Page 2)
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Online 14 - Section 7.2-Doug Ensley
ID: 7.2.18
3. Jan's All You Can Eat Restaurant charges $8.65 per customer to eat at the restaurant. Restaurant management finds that
its expense per customer, based on how much the customer eats and the expense of labor, has a distribution that is
skewed to the right with a mean of $8.20 and a standard deviation of $3. Complete parts (a) and (b).
a. If the 100 customers on a particular day have the characteristics of a random sample from their customer base, find the
mean and standard deviation of the sampling distribution of the restaurant's sample mean expense per customer.
The mean is
. (Round to the nearest hundredth as needed.)
The standard deviation is
. (Round to the nearest hundredth as needed.)
b. Find the probability that the restaurant makes a profit that day, with the sample mean expense being less than $8.65.
(Hint: Apply the central limit theorem to the sampling distribution in (a).)
(Round to the nearest thousandth as needed.)
ID: 7.2.20
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Online 14 - Section 7.2-Doug Ensley
4. A person's blood pressure is monitored by taking 4 readings daily. The probability distribution of his reading had a mean of
128 and a standard deviation of 3.
a. Each observation behaves as a random sample. Find the mean and the standard deviation of the sampling distribution of
the sample mean for the four observations each day.
b. Suppose that the probability distribution of his blood pressure reading is normal. What is the shape of the sampling
distribution?
c. Refer to (b). Find the probability that the sample mean exceeds 140.
a. The mean of the sampling distribution is
The standard deviation is
.
(Round to three decimal places as needed.)
b. Choose the correct description of the shape of the sampling distribution of x for a sample size of 4.
A. The distribution is approximately normal.
B. The distribution is skewed left.
C. The distribution is uniform.
D. The distribution is skewed right.
E. The shape of the distribution is unknown.
c. The probability that the sample mean exceeds 140 is
(Round to four decimal places as needed.)
ID: 7.2.22
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.
Online 14 - Section 7.2-Doug Ensley
5. A large corporation employs 27,456 individuals. The average income several years ago for all employees was $74,824 with
a standard deviation of $19,327. You are interested in comparing the incomes of today's employees with those of several
years ago. A random sample of 100 employees of the corporation yields x = $75,964 and s = $18,300. Complete parts a
through d below.
a. Describe the center and variability of the population distribution. What shape does it probably have? Explain.
The center of the population distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
The variability of the population distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
What shape does the population distribution probably have?
A. It is probably approximately normal because the population size is very large. Populations that
are sufficiently large are always approximately normal.
B. It is probably uniform because the company probably has many set levels of employee ability
where incomes are fixed for a given ability level.
C. It is probably skewed right because the company probably has many lower-level employees with
lower incomes and a few upper-level employees with very high incomes.
D. It is probably skewed left because the company probably has many upper-level employees with
very high incomes and a few lower-level employees with lower incomes
b. Describe the center and variability of the data distribution. What shape does it probably have? Explain.
The center of the data distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
The variability of the data distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
What shape does the data distribution probably have?
The data distribution is probably (1)
because the population distribution is probably (2)
c. Describe the center and variability of the sampling distribution of the sample mean for n = 100. What shape does it have?
Explain.
The center of the sampling distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
The variability of the sampling distribution is
.
(Type an integer or a decimal. Do not include the $ symbol in your answer.)
What shape does the sampling distribution have?
A. It is uniform because the population distribution is probably uniform.
B. It is skewed right because the population distribution is probably skewed right.
C. It is approximately normal because all sampling distributions with sufficiently large sample sizes
are approximately normal.
D. It is skewed left because the population distribution is probably skewed left.
d. Explain why it would not be unusual to observe an individual who earns more than $100,000, but it would be highly
unusual to observe a sample mean income of more than $100,000 for a random sample size of 100 people.
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Online 14 - Section 7.2-Doug Ensley
It would not be unusual to observe an individual earning more than $100,000 because this is well (3)
(4)
standard deviations of the mean. It would be highly unusual to observe a sample mean income above
$100,000 for a random sample size of 100 people because this is well (5)
deviations from the mean.
(1)
(2)
uniform
skewed right
skewed left
beyond
within
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(6)
skewed left.
approximately normal.
approximately normal
(5)
three
uniform.
skewed right.
sampling distribution
population
(3)
three (6)
beyond
within
(4)
standard
sampling distribution
population
Online 14 - Section 7.2-Doug Ensley
1.
A. The number of years of education
13.4
C. The expected value for the mean of a sample of size 49
0.7
A. The variablility of the mean for samples of size 49
13.4
0.35
(1) stays the same
(2) decreases
2.
17
B.
The empirical rule says that if a distribution of data is bell shaped, then all or nearly all observations fall within 3 standard
deviations of the mean. The mean of a sample of such observations would also fall within 3 standard deviations of the overall
mean.
0.8239
3.
8.20
0.3
0.933
4.
128
1.5
A. The distribution is approximately normal.
0
5.
74,824
19,327
C.
It is probably skewed right because the company probably has many lower-level employees with lower incomes and a few
upper-level employees with very high incomes.
75,964
18,300
(1) skewed right
(2) skewed right.
74,824
1932.7
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C.
Online 14 - Section 7.2-Doug Ensley
It is approximately normal because all sampling distributions with sufficiently large sample sizes are approximately normal.
(3) within
(4) population
(5) beyond
(6) sampling distribution
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