Student: _____________________
Date: _____________________
Instructor: Ahmad Daryanto
Course: MKTG100 Numeracy and Stats
Assignment: 2HW8B Sampling
distribution of means
Given a normal distribution with and , and given you select a sample of n , complete parts (a) through (d).μ = 102 σ = 10 = 4
a. What is the probability that is less than ?X 94
P( )X < 94 =
(Type an integer or decimal rounded to four decimal places as needed.)
b. What is the probability that is between and ?X 94 96.5
P( )94 < X < 96.5 =
(Type an integer or decimal rounded to four decimal places as needed.)
c. What is the probability that is above ?X 103.2
P( )X > 103.2 =
(Type an integer or decimal rounded to four decimal places as needed.)
d. There is a % chance that is above what value?68 X
X =
(Type an integer or decimal rounded to two decimal places as needed.)
Given a normal distribution with and , and given you select a sample of , complete parts (a) through (d).μ = 51 σ = 8 n = 100
a. What is the probability that is less than ?X 49
P( )X < 49 =
(Type an integer or decimal rounded to four decimal places as needed.)
b. What is the probability that is between and ?X 49 50.5
P( )49 < X < 50.5 =
(Type an integer or decimal rounded to four decimal places as needed.)
c. What is the probability that is above ?X 51.7
P( )X > 51.7 =
(Type an integer or decimal rounded to four decimal places as needed.)
d. There is a % chance that is above what value?40 X
X =
(Type an integer or decimal rounded to two decimal places as needed.)
3.
4.
Given a population of ,
indicate what the sampling distribution for samples of would consist of.
absentee records (days absent per year) in 2007 for employees of a large manufacturing company
20
Choose the correct answer below.
A. The sampling distribution is the average result from all possible samples of .20 records
B. The sampling distribution is a representative collection of samples, each containing
, selected without replacement.
20 20
records
C. The sampling distribution is a representative collection of samples, each containing
, selected with replacement.
20 20
records
D. The sampling distribution is the distribution of the results for all possible samples of .20 records
(1) smaller than
the same as
larger than
(2) greater
less
(3) decreases,
increases,
(4) decrease
increase
The amount of water in a bottle is approximately normally distributed with a mean of liters with a standard deviation of
liter. Complete parts (a) through (e) below.
2.45
0.035
a. What is the probability that an individual bottle contains less than liters?2.42
(Round to three decimal places as needed.)
b. If a sample of bottles is selected, what is the probability that the sample mean amount contained is less than
liters?
4 2.42
(Round to three decimal places as needed.)
c. If a sample of bottles is selected, what is the probability that the sample mean amount contained is less than
liters?
25 2.42
(Round to three decimal places as needed.)
d. Explain the difference in the results of (a) and (c).
Part (a) refers to an individual bottle, which can be thought of as a sample with sample size . Therefore, the
standard error of the mean for an individual bottle is times the standard error of the sample in (c) with
sample size 25. This leads to a probability in part (a) that is (1) the probability in part (c).
(Type integers or decimals. Do not round.)
e. Explain the difference in the results of (b) and (c).
The sample size in (c) is greater than the sample size in (b), so the standard error of the mean (or the standard deviation of
the sampling distribution) in (c) is (2) than in (b). As the standard error (3) values become
more concentrated around the mean. Therefore, the probability that the sample mean will fall close to the population mean
will always (4) when the sample size increases.
5. The diameter of a brand of tennis balls is approximately normally distributed, with a mean of inches and a standard
deviation of inch. A random sample of tennis balls is selected. Complete parts (a) through (d) below.
2.61
0.06 12
a. What is the sampling distribution of the mean?
A. Because the population diameter of tennis balls is approximately normally distributed, the
sampling distribution of samples of size will not be approximately normal.12
B. Because the population diameter of tennis balls is approximately normally distributed, the
sampling distribution of samples of size will also be approximately normal.12
C. Because the population diameter of tennis balls is approximately normally distributed, the
sampling distribution of samples of size will be the uniform distribution.12
D. Because the population diameter of tennis balls is approximately normally distributed, the
sampling distribution of samples of size cannot be found.12
b. What is the probability that the sample mean is less than inches? 2.58
P( )X < 2.58 =
(Round to four decimal places as needed.)
c. What is the probability that the sample mean is between and inches?2.59 2.63
P( )2.59 < X < 2.63 =
(Round to four decimal places as needed.)
d. The probability is % that the sample mean will be between what two values symmetrically distributed around the
population mean?
56
The lower bound is inches. The upper bound is inches.
(Round to two decimal places as needed.)
6.
(1) less
greater
(2) decreases,
increases,
(3) decrease
increase
According to a report from a business intelligence company, smartphone owners are using an average of apps per
month. Assume that number of apps used per month by smartphone owners is normally distributed and that the standard
deviation is . Complete parts (a) through (d) below.
19
7
a. If you select a random sample of smartphone owners, what is the probability that the sample mean is between
and ?
16 18.5
19.5
(Round to three decimal places as needed.)
b. If you select a random sample of smartphone owners, what is the probability that the sample mean is between and
?
16 18
19
(Round to three decimal places as needed.)
c. If you select a random sample of smartphone owners, what is the probability that the sample mean is between
and ?
100 18.5
19.5
(Round to three decimal places as needed.)
d. Explain the difference in the results of (a) and (c).
The sample size in (c) is greater than the sample size in (a), so the standard error of the mean (or the standard deviation of
the sampling distribution) in (c) is (1) than in (a). As the standard error (2) values become
more concentrated around the mean. Therefore, the probability that the sample mean will fall close to the population mean
will always (3) when the sample size increases.