8.2 Homework-Kacie Thao
12/12/22, 6:56 PM
Instructor: Patrick Mitchell
Course: MATH-1610-001 Statistics for
Decision Making - Fall 2022
Student: Kacie Thao
Date: 12/12/22
Assignment: 8.2 Homework
Construct a 99% confidence interval to estimate the population mean with x = 95 and σ = 23 for the following sample sizes.
a) n = 42
b) n = 57
c) n = 75
a) A (1 − α) × 100% confidence interval for the mean, with σ known, is given by the formula below, where x is the sample mean, zα / 2 is the
critical z-score, σx is the standard error of the mean, σ is the population standard deviation, and n is the sample size.
UCLx = x + zα / 2 σ
x
LCLx = x − zα / 2 σ
x
where σx =
σ
n
First find α.
A 99% confidence level corresponds to an α-value of 0.01.
Compute
α
2
=
α
2
.
0.01
2
= 0.005
Now compute 1 −
1−
α
2
α
2
.
= 1 − 0.005
= 0.995
Next find zα / 2 . While either technology or a cumulative standardized normal distribution table can be used to find zα / 2 , for the purposes of this
explanation, use the table, rounding to two decimal places.
zα / 2 = z0.005
= 2.58
Now compute σx, where n = 42, rounding to two decimal places.
σx =
=
σ
n
23
42
= 3.55
Construct a 99% confidence interval estimate of the population mean, μ. First compute UCLx , rounding to two decimal places.
UCLx = x + zα / 2 σ
x
= 95 + (2.58)(3.55)
= 104.16
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8.2 Homework-Kacie Thao
12/12/22, 6:56 PM
Now compute LCLx , rounding to two decimal places.
LCL
x = x − zα / 2 σx
= 95 − (2.58)(3.55)
= 85.84
Thus, with 99% confidence and n = 42, conclude that the population mean is between the lower limit of 85.84 and the upper limit of 104.16.
b) Compute the confidence interval for the mean when n = 57 by repeating the process in part a.
Start by computing σx, where n = 57, rounding to two decimal places.
σx =
=
σ
n
23
57
= 3.05
Construct a 99% confidence interval estimate of the population mean, μ. First compute UCLx , rounding to two decimal places.
UCLx = x + zα / 2 σ
x
= 95 + (2.58)(3.05)
= 102.87
Now compute LCLx , rounding to two decimal places.
LCLx = x − zα / 2 σ
x
= 95 − (2.58)(3.05)
= 87.13
Thus, with 99% confidence and n = 57, conclude that the population mean is between the lower limit of 87.13 and the upper limit of 102.87.
c) Compute the confidence interval for the mean when n = 75 by repeating the process in part a.
Start by computing σx, where n = 75, rounding to two decimal places.
σx =
=
σ
n
23
75
= 2.66
Construct a 99% confidence interval estimate of the population mean, μ. First compute UCLx , rounding to two decimal places.
UCLx = x + zα / 2 σ
x
= 95 + (2.58)(2.66)
= 101.86
Now compute LCLx , rounding to two decimal places.
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8.2 Homework-Kacie Thao
LCL
12/12/22, 6:56 PM
x = x − zα / 2 σx
= 95 − (2.58)(2.66)
= 88.14
Thus, with 99% confidence and n = 75, conclude that the population mean is between the lower limit of 88.14 and the upper limit of 101.86.
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