The chi-square distribution results when independent variables with standard normal distributions...
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![Page 1: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/1.jpg)
The chi-square distribution results when independent variables with standard normal distributions are squared and summed.
22 ( )
1ix x
sn
Sampling distribution of s2
![Page 2: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/2.jpg)
22
2-stat ( 1)
sn
The chi-square distribution results when independent variables with standard normal distributions are squared and summed.
0 n – 1
2 ( 1)n
Sampling distribution ofc2
![Page 3: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/3.jpg)
The chi-square distribution results when independent variables with standard normal distributions are squared and summed.
2.025
.025
22
2-stat ( 1)
sn
Sampling distribution ofc2
![Page 4: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/4.jpg)
The chi-square distribution results when independent variables with standard normal distributions are squared and summed.
2.975
.975
22
2-stat ( 1)
sn
Sampling distribution ofc2
![Page 5: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/5.jpg)
2 2.975 5
2
22 .0
( 1)n s
2
2.975
2( 1)n s
2
22.025
( 1)n s
Interval Estimation of s 2
To derive the interval estimate of 2, first substitute(n - 1)s2/ 2 for c2 into the following inequality
22 2.975 .025
Now write the above as two inequalities
![Page 6: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/6.jpg)
2
22.0252
2( 1)n s
2
2
( 1)n s 2 2.025
2 2 2.025
2( 1)n s
Next, multiply the inequalities by 2
22
.975 22
2( 1)n s
2
2.9
275 2
( 1)n s 2 22 2.975 ( 1)n s
Divide both of the inequalities above by the respective chi-square critical value:
2 2
.2
2 2.025 .025
025( 1)n s
2
.02
2.0
5
225( 1)n s
2
2.025
22.025
2( 1)n s
2
2 2.975 .975
22.975 ( 1)n s 2.975
2
2.975
2
2.975
( 1)n s
22.97
2
5
( 1)n s
Interval Estimation of s 2
2 2
22 2.025 .975
( 1) ( 1)n s n s
![Page 7: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/7.jpg)
Interval Estimation of s 2
2 2
22 2.025 .975
( 1) ( 1)n s n s
The 95% confidence interval for the population variance
![Page 8: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/8.jpg)
2 2
22 2.025 .975
( 1) ( 1)n s n s
Interval Estimation of s 2
1 - a/2a/2
The 1 - a confidence interval for the population variance
![Page 9: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/9.jpg)
Buyer’s Digest rates thermostats manufactured for home temperature control. In a recent test, ten thermostats manufactured by ThermoRite were selected at random and placed in a test room that was maintained at a temperature of 68oF.
Use the ten readings in the table below to develop a 95% confidence interval estimate of the population variance.
Example 1
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
Thermostat 1 2 3 4 5 6 7 8 9 10
Interval Estimation of s 2
![Page 10: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/10.jpg)
67.4 67.8 68.2 69.3 69.5 67.068.1 68.667.967.2
-0.7-0.30.11.21.4
-1.10.00.5
-0.2-0.9
0.490.090.011.441.961.210.000.250.040.81
22 ( )
1ix x
sn
2( )ix xix x
sum = 6.3s 2 = 0.7
268 )1( .ix 68.1ix ix
Interval Estimation of s 2
68.1x
![Page 11: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/11.jpg)
Interval Estimation of s 2
2 22
2 2/ 2 1 / 2
( 1) ( 1)n s n s(10 -1)(0.7) (10 -1)(0.7)
![Page 12: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/12.jpg)
Interval Estimation of s 2
2 22
2 2/ 2 1 / 2
( 1) ( 1)n s n s(10 -1)(0.7) (10 -1)(0.7)
290 .975
2
.025
.0252
.025
1-a = .95
0.9750.25
![Page 13: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/13.jpg)
Selected Values from the Chi-Square Distribution Table
Degrees Area in Upper Tail
of Freedom .99.975 .95 .90 .10 .05 .025 .01
50.55
40.83
11.14
51.61
0 9.23611.07
012.83
215.08
6
60.87
21.23
71.63
52.20
410.64
512.59
214.44
916.81
2
71.23
91.69
02.16
72.83
312.01
714.06
716.01
318.47
5
81.64
72.18
02.73
33.49
013.36
215.50
717.53
520.09
0
92.08
82.70
03.32
54.16
814.68
416.91
919.02
321.66
6
102.55
83.24
73.94
04.86
515.98
718.30
720.48
323.20
9
2.025
2.975
Interval Estimation of s 2
![Page 14: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/14.jpg)
Interval Estimation of s 2
2 22
2 2/ 2 1 / 2
( 1) ( 1)n s n s(10 -1)(0.7) (10 -1)(0.7)
0.9750.2519.023 2.700
20.331 2.333
We are 95% confident that the populationvariance is in this interval
![Page 15: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/15.jpg)
Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable” rating to a thermostat with a temperature variance of 0.5 or less.
Conduct a hypothesis test--at the 10% significance level--to
determine whether the ThermoRite thermostat’s temperature
variance is “acceptable”.
Hypotheses: 20 : 0.5H
2: 0.5aH
2 -stat
Recall that s2 = 0.7 and df = 9. With = 0.5, 20
2
20
( 1)sn
(0.(9) 7)
0.512.6
Example 2
Hypothesis Testing – One Variance
![Page 16: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/16.jpg)
a = .10 (column)
Selected Values from the Chi-Square Distribution Table
Degrees Area in Upper Tail
of Freedom .99.975 .95 .90 .10 .05 .025 .01
50.55
40.83
11.14
51.61
0 9.23611.07
012.83
215.08
6
60.87
21.23
71.63
52.20
410.64
512.59
214.44
916.81
2
71.23
91.69
02.16
72.83
312.01
714.06
716.01
318.47
5
81.64
72.18
02.73
33.49
013.36
215.50
717.53
520.09
0
92.08
82.70
03.32
54.16
814.68
416.91
919.02
321.66
6
102.55
83.24
73.94
04.86
515.98
718.30
720.48
323.20
9
Our value 2
.10
and df = 10 – 1 = 9 (row)
Hypothesis Testing – One Variance
![Page 17: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/17.jpg)
214.684
.10
2-stat
Do not reject H0 Reject H0
9
There is insufficient evidence to conclude that the temperature variance for ThermoRite thermostats is unacceptable.
= .101 9n 20 : 0.5H
12.6
Hypothesis Testing – One Variance
![Page 18: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/18.jpg)
The F-distribution results from taking the ratio of variances of normally distributed variables.
Sampling distribution of F
2122
1
if s12 = s2
2
![Page 19: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/19.jpg)
The F-distribution results from taking the ratio of variances of normally distributed variables.
Sampling distribution of F
Bigger
2122
-stats
Fs
≈1 if s12 = s2
2
0 1
![Page 20: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/20.jpg)
.025F
.025
The F-distribution results from taking the ratio of variances of normally distributed variables.
Sampling distribution of F
2122
-stats
Fs
≈1
![Page 21: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/21.jpg)
.975F
The F-distribution results from taking the ratio of variances of normally distributed variables.
Sampling distribution of F
2122
-stats
Fs
≈1
.975
![Page 22: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/22.jpg)
Buyer’s Digest has conducted the same test, but on 10 other thermostats. This time it test thermostats manufactured by TempKing. The temperature readings of the 10 thermostats are listed below.
We will conduct a hypothesis at a 10% level of significance to see if the variances are equal for both thermostats.
Example 3
ThermoRite Sample
TempKing Sample
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
Temperature 67.7 66.4 69.2 70.1 69.5 69.7 68.1 66.6 67.3 67.5
s2 = 0.7 and df = 9
s2 = ? and df = 9
Hypothesis Testing – Two Variances
![Page 23: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/23.jpg)
67.766.469.270.169.569.768.166.667.367.5
-0.51-1.810.991.891.291.49
-0.11-1.61-0.91-0.71
0.26013.27610.98013.57211.66412.22010.01212.59210.82810.5041
ix
22 ( )
1ix x
sn
2( )ix xix x
sum = 15.909s 2 = 1.768
68.21x
268 )1( .2ix 68.21ix TempKing
Since this is largerThan ThermoRite’s
21 1.768s
22 0.7s
Hypothesis Testing – Two Variances
![Page 24: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/24.jpg)
n1 = 10 – 1 = 9 (column)
Selected Values from the F Distribution Table
Denominator Area in Numerator Degrees of Freedom
Degrees Upper
of Freedom Tail 7 8 9 10 15
.01 6.18 6.03 5.91 5.81 5.52
9 .10 2.51 2.47 2.44 2.42 2.34
.05 3.29 3.23 3.18 3.14 3.01
.025 4.20 4.10 4.03 3.96 3.77
.01 5.61 5.47 5.35 5.26 4.96.05F
& n 2 - 1 = 9 a/2 = .05 (row)
Hypothesis Testing – Two Variances
Hypotheses: 2 2
0 1 2:H 2 2
1 2:aH
![Page 25: The chi-square distribution results when independent variables with standard normal distributions are squared and summed. Sampling distribution ofs2s2.](https://reader036.fdocuments.in/reader036/viewer/2022070308/551c2b975503469e4f8b609b/html5/thumbnails/25.jpg)
F.95F
.05
Reject H0 Do not Reject H0 Reject H0
≈ 1
There is insufficient evidence to conclude that the population
variances differ for the two thermostat brands.
3.18
.05
Hypothesis Testing – Two Variances
-statF2122
ss
1.7680.70
2.53
2.53