Download Solutions to the worksheet

Solutions
Descriptive Statistics
Sample 1
Mean
Standard Deviation
Sample Variance
2.2993
0.41905
0.1756
Sample 2
Mean
Standard Deviation
Sample Variance
2.29471
0.41496
0.1722
Sample 3
Mean
Standard Deviation
Sample Variance
2.25266
0.41457
0.17187
Sample 4
Mean
Standard Deviation
Sample Variance
2.17571
0.4113
0.16917
Estimating the Mean
We start by computing the sample means:
Sample
count
mean
1
100
2.2993
2
50
2.29471
3
20
2.25266
4
10
2.17571
We have the following critical z-scores for various levels of confidence:
confidence
critical z
0.8
1.28155
0.9
1.64485
0.95
1.95996
0.99
2.57583
0.93
1.81191
0.999
3.29053
Assuming that the variance is known, and equal to 0.16 (σ=0.4), we have the following confidence
intervals for the four samples (the theoretical mean was 2.3, and all intervals include it) :
confidence
left #1
right #1
left #2
right #2
left #3
right #3
left #4
right #4
0.8
2.24804
2.35056
2.22222
2.36721
2.13803
2.36728
2.01361
2.33782
0.9
2.23351
2.3651
2.20167
2.38776
2.10553
2.39978
1.96765
2.38377
0.95
2.2209
2.3777
2.18384
2.40559
2.07735
2.42796
1.92779
2.42363
0.99
2.19627
2.40234
2.149
2.44043
2.02227
2.48304
1.84989
2.50153
0.93
2.22683
2.37178
2.19222
2.39721
2.09059
2.41472
1.94652
2.4049
0.999
2.16768
2.43092
2.10857
2.48086
1.95834
2.54697
1.75949
2.59193
Estimating the Variance
Known Mean
If the mean is known, we compute the expression
the statistic that has a Â2n distribution, obtaining
Sample
n
X
2
(xk ¡ ¹)
1
17.3846
n
X
k=1
2
(xk ¡ ¹) , which goes in the numerator of
2
8.43898
3
3.31039
4
1.67696
k=1
The critical Â-scores turn out to be (denoting them by  n;L and  n;R )
Confidence
chi_100,L
chi_100,R
chi_50,L
chi_50,R
chi_20,L
chi_20,R
chi_10,L
chi_10,R
0.9
124.342
77.9295
67.5048
34.7643
31.4104
10.8508
18.307
3.9403
0.92
126.079
76.6705
68.8039
33.9426
32.3206
10.4154
19.0207
3.69654
0.95
129.561
74.2219
71.4202
32.3574
34.1696
9.59078
20.4832
3.24697
0.98
135.807
70.0649
76.1539
29.7067
37.5662
8.2604
23.2093
2.55821
0.99
140.169
67.3276
79.49
27.9907
39.9968
7.43384
25.1882
2.15586
Applying the information above to the formula for the confidence interval for ¾ 2 results in the
following
known mean
left #1
right #1
left #2
right #2
left #3
right #3
left #4
right #4
0.9
0.13981
0.22308
0.12501
0.24275
0.10539
0.30508
0.0916
0.42559
0.92
0.13789
0.22674
0.12265
0.24863
0.10242
0.31784
0.08817
0.45366
0.95
0.13418
0.23422
0.11816
0.26081
0.09688
0.34516
0.08187
0.51647
0.98
0.12801
0.24812
0.11081
0.28408
0.08812
0.40075
0.07225
0.65552
0.99
0.12403
0.25821
0.10616
0.30149
0.08277
0.44531
0.06658
0.77786
Unknown Mean
When the mean is unknown, the numerator in the statistic we use (which will now have a  2n¡1
n
X
2
(xk ¡ x¹ ) = (n ¡ 1)s2. The last expression means to take advantage
distribution) is given by
k=1
of the sample variance, which we calculated at the beginning.
Sample
(n ¡ 1)s2
1
17.3846
2
8.43758
3
3.26556
4
1.52249
The critical Â-scores are now (denoted by  2(n¡1);L and  2(n¡1);R)
Confidence
chi_99,L
chi_99,R
chi_49,L
chi_49,R
chi_19,L
chi_19,R
chi_9,L
chi,9,R
0.9
123.225
77.0463
66.3386
33.9303
30.1435
3.32511
16.919
3.32511
0.92
124.955
75.7949
67.6271
33.1192
31.0367
3.10467
17.6083
3.10467
0.95
128.422
73.3611
70.2224
31.5549
32.8523
2.70039
19.0228
2.70039
0.98
134.642
69.2299
74.9195
28.9406
36.1909
2.0879
21.666
2.0879
0.99
138.987
66.5101
78.2307
27.2493
38.5823
1.73493
23.5894
1.73493
0.95
0.13537
0.23697
0.12016
0.26739
0.0994
1.20929
0.08004
0.5638
0.98
0.12912
0.25111
0.11262
0.29155
0.09023
1.56404
0.07027
0.7292
0.99
0.12508
0.26138
0.10786
0.30964
0.08464
1.88224
0.06454
0.87755
resulting in the following confidence intervals
Unknown mean
left #1
right #1
left #2
right #2
left #3
right #3
left #4
right #4
0.9
0.14108
0.22564
0.12719
0.24867
0.10833
0.98209
0.08999
0.45788
0.92
0.13913
0.22936
0.12477
0.25476
0.10522
1.05182
0.08646
0.49039
As the sample size gets smaller and/or the confidence level rises, the width of the interval
becomes larger, sometimes very much larger. Fortunately, the “true” variance (the one used in
simulating the data) of .16 falls within all intervals.