Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
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Chapter 8.5, Problem 24E
To determine

To build and to interpret:

A 98% confidence interval for the population variance for the weights of all boxes of crackers that come off the assembly line.

Expert Solution & Answer
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Answer to Problem 24E

Solution:

The confidence interval for the population variance is given by (0.0019, 0.0128).

Thus, we are 98% confident that the population variance is between 0.0019 and 0.0128.

Explanation of Solution

Procedure:

Steps need to be followed while calculating the confidence interval.

STEP 1:

Find the point estimate, s2 for confidence interval for the population variance and s for confidence interval for the population standard deviation.

STEP 2:

Calculate σ2 and 1-σ2, based on the level of confidence c given.

STEP 3:

Find the critical value χσ22 and χ1-σ22 for the distribution with n-1 degrees of freedom using the χ2-distribution table.

STEP 4:

Find the confidence interval for the population variance by substituting the necessary values in the formula

n-1s2χσ22<σ2<n-1s2χ1-σ22

Find the confidence interval for the population standard deviation by substituting the necessary values in the formula

n-1s2χσ22<σ<n-1s2χ1-σ22

Calculation:

Level confidence =98%

First let us calculate the sample variance for the given data.

The sample variance of a data having ‘n’ number of data values in the sample with mean ‘x-’ is given by

s2=xi-x-2n-1

Here, we need to find the mean ‘x-

x-=xin

=16.87+16.92+17.01+16.98+16.99+16.92+16.91+17.00+                                    17.01+16.96+16.95+16.94+17.00+16.9214

x-=237.3814

x-=16.9557

x-=17

Now, construct a table of deviations and squared deviations of the data.

Deviations and squared Deviations of the data
xi xi-x¯ (xi-x¯)2
16.87 16.87-17=-0.13 0.0169
16.92 16.92-17=-0.08 0.0064
17.01 17.01-17=0.01 0.0001
16.98 16.98-17=-0.02 0.0004
16.99 16.99-17=-0.01 0.0001
16.92 16.92-17=-0.08 0.0064
16.91 16.91-17=-0.09 0.0081
17.00 17.00-17=0 0
17.01 17.01-17=0.01 0.0001
16.96 16.96-17=-0.04 0.0016
16.95 16.95-17=-0.05 0.0025
16.94 16.94-17=-0.06 0.0036
17.00 17.00-17=0 0
16.92 16.92-17=-0.08 0.0064

(xi-x¯)2=0.0526

Thus, we have

(xi-x¯)2=0.0526 (Squared deviation)

n=14 (Sample size)

Substituting the above values in

s2=xi-x-2n-1

     =0.052614-1

     =0.052613

s2=0.00405

Now, construct a confidence interval for the population variance with

n=14 (Sample size)

s2=0.00405 (Sample variance)

c=0.98 (Level of confidence)

STEP 1:

Find the point estimate:

Here, we need to construct the confidence interval for the population variance, thus the point estimate is the value of s2 and is given as s2=0.00405

STEP 2:

Calculate σ2 and 1-σ2, based on the level of confidence given, for the level of confidence given,

c=0.98

We have,

σ=1-c

σ=1-0.98

σ=0.02

Thus,

=σ2

=0.022

=0.01 and

1-σ2

=1-0.01

=0.99

STEP 3:

Find the critical value χσ22 and χ1-σ22 for the distribution with n-1 degrees of freedom using the χ2-distribution table,

=χσ22

=χ0.012

=27.69 and

χ1-σ22

=χ1-0.012

=χ0.992

=4.107

STEP 4:

Find the confidence interval by substituting the necessary values in the formula:

n-1s2χσ22<σ2<n-1s2χ1-σ22

14-10.0040527.69<σ2<14-10.004054.107

130.0040527.69<σ2<130.004054.107

0.0526527.69<σ2<0.052654.107

0.0019<σ2<0.0128

Using, interval notation, the confidence interval can also be written as (0.0019, 0.0128)

Final statement:

Therefore, the confidence interval for the population variance is given by (0.0019, 0.0128).

Thus, we are 98% confident that the population variance is between 0.0019 and 0.0128.

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Chapter 8 Solutions

Beginning Statistics, 2nd Edition

Ch. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - Prob. 14ECh. 8.1 - Prob. 15ECh. 8.1 - Prob. 16ECh. 8.1 - Prob. 17ECh. 8.1 - Prob. 18ECh. 8.1 - Prob. 19ECh. 8.1 - Prob. 20ECh. 8.1 - Prob. 21ECh. 8.1 - Prob. 22ECh. 8.1 - Prob. 23ECh. 8.1 - Prob. 24ECh. 8.1 - Prob. 25ECh. 8.1 - Prob. 26ECh. 8.1 - Prob. 27ECh. 8.1 - Prob. 28ECh. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - Prob. 31ECh. 8.1 - Prob. 32ECh. 8.2 - Prob. 1ECh. 8.2 - Prob. 2ECh. 8.2 - Prob. 3ECh. 8.2 - Prob. 4ECh. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - Prob. 8ECh. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - Prob. 11ECh. 8.2 - Prob. 12ECh. 8.2 - Prob. 13ECh. 8.2 - Prob. 14ECh. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.2 - Prob. 18ECh. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - Prob. 22ECh. 8.2 - Prob. 23ECh. 8.2 - Prob. 24ECh. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.3 - Prob. 1ECh. 8.3 - Prob. 2ECh. 8.3 - Prob. 3ECh. 8.3 - Prob. 4ECh. 8.3 - Prob. 5ECh. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Prob. 8ECh. 8.3 - Prob. 9ECh. 8.3 - Prob. 10ECh. 8.3 - Prob. 11ECh. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Prob. 14ECh. 8.3 - Prob. 15ECh. 8.3 - Prob. 16ECh. 8.3 - Prob. 17ECh. 8.3 - Prob. 18ECh. 8.3 - Prob. 19ECh. 8.3 - Prob. 20ECh. 8.3 - Prob. 21ECh. 8.3 - Prob. 22ECh. 8.3 - Prob. 23ECh. 8.3 - Prob. 24ECh. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.4 - Prob. 1ECh. 8.4 - Prob. 2ECh. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 7ECh. 8.4 - Prob. 8ECh. 8.4 - Prob. 9ECh. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Prob. 13ECh. 8.4 - Prob. 14ECh. 8.4 - Prob. 15ECh. 8.4 - Prob. 16ECh. 8.4 - Prob. 17ECh. 8.4 - Prob. 18ECh. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Prob. 21ECh. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.5 - Prob. 6ECh. 8.5 - Prob. 7ECh. 8.5 - Prob. 8ECh. 8.5 - Prob. 9ECh. 8.5 - Prob. 10ECh. 8.5 - Prob. 11ECh. 8.5 - Prob. 12ECh. 8.5 - Prob. 13ECh. 8.5 - Prob. 14ECh. 8.5 - Prob. 15ECh. 8.5 - Prob. 16ECh. 8.5 - Prob. 17ECh. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - Prob. 20ECh. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Prob. 25ECh. 8.5 - Prob. 26ECh. 8.5 - Prob. 27ECh. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.CR - Prob. 1CRCh. 8.CR - Prob. 2CRCh. 8.CR - Prob. 3CRCh. 8.CR - Prob. 4CRCh. 8.CR - Prob. 5CRCh. 8.CR - Prob. 6CRCh. 8.CR - Prob. 7CRCh. 8.CR - Prob. 8CRCh. 8.CR - Prob. 9CRCh. 8.CR - Prob. 10CRCh. 8.CR - Prob. 11CRCh. 8.CR - Prob. 12CRCh. 8.CR - Prob. 13CRCh. 8.PA - Prob. 1PCh. 8.PA - Prob. 2PCh. 8.PA - Prob. 3PCh. 8.PA - Prob. 4PCh. 8.PA - Prob. 5PCh. 8.PB - Prob. 1PCh. 8.PB - Prob. 2PCh. 8.PB - Prob. 3PCh. 8.PB - Prob. 4PCh. 8.PB - Prob. 5P
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