252onesx3 1/15/08
Confidence Intervals and Hypothesis tests for a Variance or Standard
Deviation including situations where the Distribution is not Normal
In Parts A and B of this course we did confidence intervals and hypothesis tests for variances. Parts a) and
b) below were done for you in the section on confidence intervals for standard deviations or variances.
Parts c) and d) were done for you in the section on hypothesis tests for a single mean or variance. Here all
the problems are presented with the underlying data and the Minitab instruction ‘OneVariance’ is used with
the subcommand ‘test’, which indicates that there is a test for a variance or standard deviation and the
subcommand ‘StDev’, which signals tests for standard deviations rather than variances. An explanation is
available from Minitab at OneVariance or on the website at
http://courses.wcupa.edu/rbove/eco252/OneVariance.pdf
a) Find a 95% confidence interval for the standard deviation if n  31 and s  1000 . We found
799    1337
b) Find a 95% confidence interval for the standard deviation if n  400 and s  1000 . We found
935    1075
In a) and b) I assumed that x  1000 . This has no effect on the results.
c) Test the hypotheses H 0 :  2  64 and H 1 :  2  64 , when s 2  100 (or s  10 ) and n  17 . We did
not reject the null hypothesis at the 2% significance level.
d) Test the hypotheses H 0 :  2  64 and H 1 :  2  64 , when s 2  100 (or s  10 ) and n  73 . We
found pval  2Pz  3.04  2.5  .4988   .0024 . Since this is below .02, we reject the null hypothesis at
the 2% significance level.
In c) and d) I assumed that x  26 . This has no effect on the results.
The results are labeled ‘standard’ if the test assumes the Normal distribution and ‘adjusted’ if the Normal
distribution is not assumed Normal. The instruction ‘describe’ is run to verify that the data sets have the
sample variances given in the problems. Names on the columns indicate the standard deviation and sample
size. The ‘adjusted’ versions will not be done if only the sample variance or standard deviation is given
instead of the data. Since one of the data columns has 400 items in it, the data goes on forever.
————— 1/15/2008 5:21:42 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My
Documents\Minitab\252vardat.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252vardat.MTW'
Worksheet was saved on Tue Jan 15 2008
Results for: 252vardat.MTW
MTB > describe c1-c4
Descriptive Statistics: 1000-31, 1000-400, 10-17, 10-73
Variable
1000-31
1000-400
10-17
10-73
N
31
400
17
73
N*
0
0
0
0
Variable
1000-31
1000-400
10-17
10-73
Maximum
4596
6258.4
43.63
55.41
Mean
3000
3000.0
26.00
26.00
SE Mean
180
50.0
2.43
1.17
StDev
1000
1000.0
10.00
10.00
Minimum
181
480.4
12.02
11.28
Q1
2509
2303.9
17.62
17.66
Median
3182
2989.5
23.23
25.48
Q3
3857
3701.4
34.38
31.38
252onesx3 1/15/08
MTB > OneVariance c1;
SUBC> StDev.
Test and CI for One Standard Deviation: 1000-31
Method
The standard method is only for the normal distribution.
The adjusted method is for any continuous distribution.
Statistics
Variable
N StDev Variance
1000-31
31
1000
1000876
95% Confidence Intervals
CI for
Variable Method
StDev
1000-31
Standard (799, 1337)
Adjusted (770, 1428)
CI for Variance The ‘standard’ interval is the
(639139, 1788260) same as was computed in the
(593343, 2038131) example. The ‘adjusted’ interval
is slightly larger.
MTB > OneVariance c2.
Test and CI for One Standard Deviation: 1000-400
Method
The standard method is only for the normal distribution.
The adjusted method is for any continuous distribution.
Statistics
Variable
N StDev Variance
1000-400 400
1000
999978
95% Confidence Intervals
CI for
Variable Method
StDev
1000-400 Standard (935, 1075)
Adjusted (938, 1071)
CI for Variance The ‘standard’ interval is the
(874528, 1154652) same as was computed in the
(879821, 1146687) example. The ‘adjusted’ interval
is slightly larger.
MTB > Onevariance c3;
SUBC> test 8.0;
SUBC> StDev.
Test and CI for One Standard Deviation: 10-17
Method
Null hypothesis
Alternative hypothesis
Sigma = 8
Sigma not = 8
The standard method is only for the normal distribution.
The adjusted method is for any continuous distribution.
Statistics
Variable
N StDev Variance
10-17
17
10.0
100
95% Confidence Intervals
CI for
Variable Method
StDev
10-17
Standard (7.4, 15.2)
Adjusted (7.9, 13.5)
Tests
Variable
10-17
Method
Standard
Adjusted
Chi-Square
25.01
44.10
CI for
Variance
(55, 232)
(63, 183)
DF
16.00
28.21
P-Value
0.139
0.058
Since the pvalue is above the
significance level(2%), we cannot
reject the null hypothesis.
252onesx3 1/15/08
MTB > OneVariance c4;
SUBC> test 8.0;
SUBC> StDev.
Test and CI for One Standard Deviation: 10-73
Method
Null hypothesis
Alternative hypothesis
Sigma = 8
Sigma not = 8
The standard method is only for the normal distribution.
The adjusted method is for any continuous distribution.
Statistics
Variable
N
10-73
73
StDev
10.0
Variance
100
95% Confidence Intervals
CI for
Variable Method
StDev
10-73
Standard (8.6, 11.9)
Adjusted (8.4, 12.4)
Tests
Variable
10-73
Method
Standard
Adjusted
Chi-Square
112.48
77.59
CI for
Variance
(74, 143)
(70, 155)
DF
72.00
49.66
P-Value
0.003
0.014
Since the pvalue is below the
significance level(2%), we can
reject the null hypothesis.
MTB > print c1-c4
Data Display
Row 1000-31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
3585.88
4208.84
3584.18
2537.54
2694.04
2751.20
3744.45
1346.52
1067.43
3975.77
3889.84
3181.92
3255.15
2509.42
1756.47
3139.21
2704.48
4092.88
2403.32
2154.06
2342.59
3857.39
3402.92
3198.86
4149.09
3867.03
2525.65
2780.30
3525.85
181.02
4596.36
1000-400 10-17 10-73
2332.38
3631.72
3728.59
3484.29
2976.47
2736.88
3177.08
4499.26
3852.93
2995.52
2702.48
4043.66
1706.15
2769.16
3352.77
3274.47
3232.55
3356.36
4251.93
3661.79
1221.20
4957.76
4115.83
3489.31
2722.16
3859.90
4058.14
2494.56
2507.21
4110.37
1645.96
4100.41
3913.43
3936.15
1484.80
2076.94
4488.63
3597.14
23.23
21.22
19.12
32.08
20.80
14.75
41.99
25.02
36.69
16.12
27.35
43.63
22.55
40.27
12.02
30.16
14.92
25.32
26.60
48.87
26.86
16.01
20.56
18.16
28.80
37.07
35.81
51.45
55.41
27.90
44.29
26.52
18.06
27.38
27.24
17.26
32.19
32.46
25.34
37.29
13.66
16.88
23.46
15.33
37.43
22.11
16.95
33.48
28.39
25.73
34.30
30.25
36.20
13.29
11.28
Remainder of printout is only on website.
252onesx3 1/15/08
39
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109
1774.88
3251.67
2221.52
1912.87
1960.33
2350.21
3052.26
4024.03
3041.15
2586.67
2824.09
4162.63
3166.53
1996.34
3348.13
4557.73
4081.44
3282.24
3103.18
3654.16
1601.42
2477.06
3286.22
4721.80
1979.27
3133.21
3551.79
4226.53
2870.94
3604.24
2469.79
4647.54
2770.07
3357.55
2851.20
1863.61
4041.75
3931.55
2335.68
2515.66
2753.84
3327.70
2812.00
2772.66
3960.05
2741.96
2847.82
2592.07
4421.32
2696.11
2665.06
3042.39
2740.32
4480.85
977.65
3744.12
2423.26
3285.62
4763.85
2792.37
3929.38
2251.02
3820.15
2102.19
910.32
1411.93
2764.54
4507.08
5915.88
2534.89
3983.83
20.56
29.25
27.26
30.57
15.78
12.83
35.71
25.48
36.25
22.54
16.95
21.66
25.50
22.44
15.38
16.41
23.71
27.87
29.55
11.41
16.63
28.46
23.77
14.16
18.09
32.20
18.83
13.67
22.85
27.71
16.22
39.64
54.42
18.72
19.91
252onesx3 1/15/08
110
111
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176
177
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179
180
1770.19
2964.52
2765.96
1805.81
2949.55
2481.94
3033.35
2800.88
4434.94
2383.93
2071.61
3530.37
1091.78
3479.44
3433.80
532.12
2344.79
2186.33
2016.65
3212.81
2625.35
3843.67
3949.20
3483.37
2602.04
1984.20
2348.46
2110.39
2596.92
3176.62
4218.21
2695.17
2495.89
1197.32
2771.44
2837.61
3086.60
2334.37
3309.69
1601.12
2294.44
2561.26
3506.28
2812.36
2181.63
2141.92
2842.33
2889.61
3028.59
3240.51
2935.72
3722.19
2819.42
3921.63
2929.52
1537.71
2695.97
3554.81
1282.16
3190.07
1660.83
1590.22
2135.77
3739.28
4411.74
3575.15
3673.32
3697.54
5055.17
1898.51
4716.79
252onesx3 1/15/08
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232
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234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
1591.77
4201.63
3094.67
2110.40
4911.63
3395.22
2806.96
4357.49
1732.98
2616.00
2547.66
2966.70
1581.03
3440.20
4249.13
3722.63
3455.32
5299.88
3090.00
4755.18
1742.58
2501.03
953.23
3991.53
2979.82
2737.08
2136.72
2757.17
1986.16
2885.58
1923.80
2934.41
4758.51
3478.32
2903.88
3674.27
2144.84
2242.74
1971.98
1030.34
2110.35
4453.39
3629.44
2979.78
4228.80
6258.45
3720.46
3026.36
2702.80
2651.12
2149.83
4588.30
3002.66
4770.34
4654.46
4418.60
3011.57
2431.78
2987.65
2522.72
1932.74
3669.19
3517.89
3125.30
1535.63
3118.88
3130.77
1580.65
2022.97
3228.78
4163.11
252onesx3 1/15/08
252
253
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305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
2848.41
1958.37
2588.27
2340.48
4143.81
3611.96
3507.27
4686.37
1661.93
4089.09
1625.85
3804.53
977.44
3163.20
998.17
3267.30
3584.50
3974.51
4533.53
2398.18
1963.06
3736.85
4074.85
2389.99
2128.68
2869.89
3098.42
2211.61
480.43
1570.77
4250.99
1029.87
1231.13
3898.61
3114.58
1258.22
2991.28
3289.39
4280.63
2662.10
2745.91
4494.05
5655.68
2212.03
1837.98
2208.59
2202.20
1977.30
2435.02
2592.33
2634.57
3231.73
4691.36
2749.36
3332.67
3223.76
4222.82
4761.06
2997.92
3340.95
4326.20
3686.37
3936.70
4381.16
1150.20
3020.05
1941.54
1991.57
3775.72
2671.23
4254.46
252onesx3 1/15/08
323
324
325
326
327
328
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331
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333
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2927.84
3519.61
3459.66
1157.49
4089.27
4853.39
3679.57
3584.13
2389.36
3092.55
3964.82
2573.31
3155.64
2929.04
3702.63
3730.52
2507.85
2004.36
2844.88
4480.43
3687.82
2859.14
1070.79
3586.39
3779.05
3294.23
3469.96
3107.17
1054.63
3587.60
1248.57
2405.29
1292.80
3719.25
4006.10
3965.97
2253.73
852.12
2601.43
1668.03
1541.09
3005.08
2372.21
1662.77
3456.84
3262.28
4515.33
3747.91
1875.45
2792.30
2293.81
709.39
3520.51
2063.99
2142.87
4856.27
2250.02
1904.11
2686.53
1209.78
3722.23
2996.36
3230.56
3102.23
2825.09
3493.37
4935.61
2161.09
2930.42
3767.48
2209.33
252onesx3 1/15/08
394
395
396
397
398
399
400
4223.12
2673.24
3488.98
3886.65
1655.96
3541.90
3273.69