Solved by verified expert:1. A furniture company wants to test a random sample of sofas to determine how long the cushions last. They simulate people sitting on the sofas by dropping a heavy object on the cushions until they wear out; they count the number of drops it takes. This test involves 9 sofas.Mean = 12,648.889s = 1,898.673Assume it follows a normal distribution. Generate a 95% confidence interval for this problem. Remember to use a t-value. Label all the parts.2. The data below are a random sample of 30 observations drawn from a population that is distributed normally with μ = 75 and σ = 8. Sampling theory tells us that most of the confidence intervals constructed from samples drawn randomly from a population contain the true population value. Check this with this sample, since we know the true population value.Mean = 76.01Std Dev = 6.48n = 30.00Construct a 95% confidence interval around the mean using σ = 8. Since sigma is known, use a z-value from the standard normal distribution. For the confidence interval, show the estimate, the z-value, the bound of error (BOE), and the lower and upper values of the confidence interval.Describe the confidence interval in words. Did the interval contain the population value of 75?Construct the same 95% confidence interval, but use the sample estimate of the standard deviation and a t-value to construct the confidence interval. Compare this result with the interval constructed in the first part.
diff.xlsx
single.xlsx
se456.pdf
se45.pdf
se4.pdf
Unformatted Attachment Preview
Excel Difference of Means Worksheet
Developed by Dr. Tom Ilvento
Values in red need to be entered for the problem
Private
n
mean
Std Dev
Var
19
78.053
9.84
96.826
Public
Ho: Difference
17 Confidence Level
77.588
Alpha for Test
8.193
67.125 Difference in Means
0
95
0.05
F-Test For Equal Varainces
The Null Hypothesis for the F-Test is that the Variances
are Equal. A high p-value implies equal variances
Ratio of Variances 1.44
F-Test
0.233
Conclusion: Assume Equal Variances
0.465
Assuming Equal Variances
Not Assuming Equal Variances
Pooled Var
Pooled Std
82.849
9.102
There is no reason to assume the variances are equal. Therefore
Standard Error for test
degrees of freedom
3.0387
34
Standard Error for test
degrees of freedom
0.4650
3.039
Hypothesis Test
Hypothesis Test
Critical Value
1.691
2.032
Confidence Interval
t-value
BOE
-5.7105
Difference
Std Error
t* =
p-value
1-tailed
2-tailed
pool the variances and the degrees of freedom are calculated diffe
Difference
Std Error
0.153
0.4396
0.8793
95 % Level
2.032
6.1755
to
6.6405
Conclusion
Fail To Reject H0
Fail To Reject H0
3.007
34
Critical Value
1.691
2.032
Confidence Interval
t-value
BOE
-5.647
t* =
p-value
1-tailed
2-tailed
95
2.032
6.112
to
al Varainces
the F-Test is that the Variances
ue implies equal variances
sume Equal Variances
ariances
variances are equal. Therefore we do not
s of freedom are calculated differently.
0.465
3.007
0.155
0.4390
0.8780
% Level
6.577
Conclusion
Fail to Reject H0
Fail to Reject H0
Difference of Proportion Test
Developed by Dr. Tom Ilvento
Values in red need to be entered for the problem
X (count)
n
p
q
Variance
Std Dev
Pintrest LinkedIn
154
171
405
379
0.3802
0.4512
0.6198
0.5488
0.2357
0.2476
0.4854
0.4976
H0: Difference =
Confidence Level
Pooled Calculations
pooled p 0.4145
Variance
pooled q 0.5855
Std Dev
If we don’t assume p1 and p2 are equal under H0:
Standard Error for test
Hypothesis Test
Confidence Interval
Z-value
BOE
-0.1398
0
95
0.0351
0.2427
0.4926
If we assume p1 and p2 are equal under H0:
Standard Error for test
0.0352
Difference
Std Error
-0.0709
0.0351
Hypothesis Test Difference
Std Error
-0.0709
0.0352
z* =
p-value
1-tailed
2-tailed
-2.0185
z* =
p-value
1-tailed
2-tailed
-2.0149
95
0.0218
0.0435
% Level
1.960
0.0689
to
-0.0021
0.0220
0.0439
0.613545817
0.822115385
1.339941309
Standard Normal Curve Probability Distribution
The table is based on the upper right 1/2 of the Normal Distribution; total area shown is .5
The Z-score values are represented by the column value + row value, up to two decimal places
The probabilities up to the Z-score are in the cells
Z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.2257
0.2580
0.2881
0.3159
0.3413
0.01
0.0040
0.0438
0.0832
0.1217
0.1591
0.1950
0.2291
0.2611
0.2910
0.3186
0.3438
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.2324
0.2642
0.2939
0.3212
0.3461
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.2357
0.2673
0.2967
0.3238
0.3485
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.2389
0.2704
0.2995
0.3264
0.3508
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
0.2422
0.2734
0.3023
0.3289
0.3531
0.06
0.0239
0.0636
0.1026
0.1406
0.1772
0.2123
0.2454
0.2764
0.3051
0.3315
0.3554
0.07
0.0279
0.0675
0.1064
0.1443
0.1808
0.2157
0.2486
0.2794
0.3078
0.3340
0.3577
0.08
0.0319
0.0714
0.1103
0.1480
0.1844
0.2190
0.2517
0.2823
0.3106
0.3365
0.3599
0.09
0.0359
0.0753
0.1141
0.1517
0.1879
0.2224
0.2549
0.2852
0.3133
0.3389
0.3621
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0.3643
0.3849
0.4032
0.4192
0.4332
0.4452
0.4554
0.4641
0.4713
0.4772
0.3665
0.3869
0.4049
0.4207
0.4345
0.4463
0.4564
0.4649
0.4719
0.4778
0.3686
0.3888
0.4066
0.4222
0.4357
0.4474
0.4573
0.4656
0.4726
0.4783
0.3708
0.3907
0.4082
0.4236
0.4370
0.4484
0.4582
0.4664
0.4732
0.4788
0.3729
0.3925
0.4099
0.4251
0.4382
0.4495
0.4591
0.4671
0.4738
0.4793
0.3749
0.3944
0.4115
0.4265
0.4394
0.4505
0.4599
0.4678
0.4744
0.4798
0.3770
0.3962
0.4131
0.4279
0.4406
0.4515
0.4608
0.4686
0.4750
0.4803
0.3790
0.3980
0.4147
0.4292
0.4418
0.4525
0.4616
0.4693
0.4756
0.4808
0.3810
0.3997
0.4162
0.4306
0.4429
0.4535
0.4625
0.4699
0.4761
0.4812
0.3830
0.4015
0.4177
0.4319
0.4441
0.4545
0.4633
0.4706
0.4767
0.4817
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.4821
0.4861
0.4893
0.4918
0.4938
0.4953
0.4965
0.4974
0.4981
0.4987
0.4826
0.4864
0.4896
0.4920
0.4940
0.4955
0.4966
0.4975
0.4982
0.4987
0.4830
0.4868
0.4898
0.4922
0.4941
0.4956
0.4967
0.4976
0.4982
0.4987
0.4834
0.4871
0.4901
0.4925
0.4943
0.4957
0.4968
0.4977
0.4983
0.4987
0.4838
0.4875
0.4904
0.4927
0.4945
0.4959
0.4969
0.4977
0.4984
0.4987
0.4842
0.4878
0.4906
0.4929
0.4946
0.4960
0.4970
0.4978
0.4984
0.4987
0.4846
0.4881
0.4909
0.4931
0.4948
0.4961
0.4971
0.4979
0.4985
0.4988
0.4850
0.4884
0.4911
0.4932
0.4949
0.4962
0.4972
0.4979
0.4985
0.4988
0.4854
0.4887
0.4913
0.4934
0.4951
0.4963
0.4973
0.4980
0.4986
0.4988
0.4857
0.4890
0.4916
0.4936
0.4952
0.4964
0.4974
0.4981
0.4986
0.4988
3.1
0.4990
0.4991
0.4991
0.4991
0.4992
0.4992
0.4992
0.4992
0.4993
0.4993
Rejection regions for Common Values of Alpha
Alternative Hypothesis
Lower Tailed Upper Tailed
Two Tailed
alpha = .10
z < -1.28
z > 1.28
z < -1.645 or z > 1.645
alpha = .05
z < -1.645
z > 1.645
z < -1.96 or z > 1.96
alpha = .01
z < -2.33
z > 2.33
z < -2.575 or z > 2.575
Prepared by Tom Ilvento using statistical functions in Excel version 2008
Critical Values of t-Distribution
The table shows the critical t-values for a given alpha level (one-tailed or two-tailed)
and degrees of freedom. The degrees of freedom are the rows.
d.f.
0.2000
0.1000
Area in Two Tails
0.0500
0.0200
0.0100
0.0020
0.0010
d.f.
0.1000
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.309
1.309
1.308
1.307
1.306
1.306
1.305
1.304
1.304
1.303
1.303
1.302
1.302
1.301
1.301
1.300
0.0500
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.696
1.694
1.692
1.691
1.690
1.688
1.687
1.686
1.685
1.684
1.683
1.682
1.681
1.680
1.679
1.679
Area in One Tail
0.0250
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.040
2.037
2.035
2.032
2.030
2.028
2.026
2.024
2.023
2.021
2.020
2.018
2.017
2.015
2.014
2.013
0.0100
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.453
2.449
2.445
2.441
2.438
2.434
2.431
2.429
2.426
2.423
2.421
2.418
2.416
2.414
2.412
2.410
0.0050
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.744
2.738
2.733
2.728
2.724
2.719
2.715
2.712
2.708
2.704
2.701
2.698
2.695
2.692
2.690
2.687
0.0010
318.309
22.327
10.215
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.610
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.375
3.365
3.356
3.348
3.340
3.333
3.326
3.319
3.313
3.307
3.301
3.296
3.291
3.286
3.281
3.277
0.0005
636.619
31.599
12.924
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.633
3.622
3.611
3.601
3.591
3.582
3.574
3.566
3.558
3.551
3.544
3.538
3.532
3.526
3.520
3.515
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
39
40
41
42
43
44
45
46
47
48
49
50
Infinity
d.f.
d.f.
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
1.300
1.299
1.299
1.299
1.282
1.678
1.677
1.677
1.676
1.645
0.2000
0.1000
0.1000
1.298
1.298
1.298
1.297
1.297
1.297
1.297
1.296
1.296
1.296
1.296
1.295
1.295
1.295
1.295
1.295
1.294
1.294
1.294
1.294
1.294
1.293
1.293
1.293
1.293
1.293
1.293
1.292
1.292
1.292
1.292
1.292
1.292
1.292
1.292
1.291
1.291
1.291
1.291
1.291
1.291
1.291
1.291
1.291
1.291
1.290
0.0500
1.675
1.675
1.674
1.674
1.673
1.673
1.672
1.672
1.671
1.671
1.670
1.670
1.669
1.669
1.669
1.668
1.668
1.668
1.667
1.667
1.667
1.666
1.666
1.666
1.665
1.665
1.665
1.665
1.664
1.664
1.664
1.664
1.663
1.663
1.663
1.663
1.663
1.662
1.662
1.662
1.662
1.662
1.661
1.661
1.661
1.661
2.012
2.011
2.010
2.009
1.960
Area in Two Tails
0.0500
Area in One Tail
0.0250
2.008
2.007
2.006
2.005
2.004
2.003
2.002
2.002
2.001
2.000
2.000
1.999
1.998
1.998
1.997
1.997
1.996
1.995
1.995
1.994
1.994
1.993
1.993
1.993
1.992
1.992
1.991
1.991
1.990
1.990
1.990
1.989
1.989
1.989
1.988
1.988
1.988
1.987
1.987
1.987
1.986
1.986
1.986
1.986
1.985
1.985
2.408
2.407
2.405
2.403
2.326
2.685
2.682
2.680
2.678
2.576
3.273
3.269
3.265
3.261
3.090
3.510
3.505
3.500
3.496
3.291
0.0200
0.0100
0.0020
0.0010
0.0100
2.402
2.400
2.399
2.397
2.396
2.395
2.394
2.392
2.391
2.390
2.389
2.388
2.387
2.386
2.385
2.384
2.383
2.382
2.382
2.381
2.380
2.379
2.379
2.378
2.377
2.376
2.376
2.375
2.374
2.374
2.373
2.373
2.372
2.372
2.371
2.370
2.370
2.369
2.369
2.368
2.368
2.368
2.367
2.367
2.366
2.366
0.0050
2.676
2.674
2.672
2.670
2.668
2.667
2.665
2.663
2.662
2.660
2.659
2.657
2.656
2.655
2.654
2.652
2.651
2.650
2.649
2.648
2.647
2.646
2.645
2.644
2.643
2.642
2.641
2.640
2.640
2.639
2.638
2.637
2.636
2.636
2.635
2.634
2.634
2.633
2.632
2.632
2.631
2.630
2.630
2.629
2.629
2.628
0.0010
3.258
3.255
3.251
3.248
3.245
3.242
3.239
3.237
3.234
3.232
3.229
3.227
3.225
3.223
3.220
3.218
3.216
3.214
3.213
3.211
3.209
3.207
3.206
3.204
3.202
3.201
3.199
3.198
3.197
3.195
3.194
3.193
3.191
3.190
3.189
3.188
3.187
3.185
3.184
3.183
3.182
3.181
3.180
3.179
3.178
3.177
0.0005
3.492
3.488
3.484
3.480
3.476
3.473
3.470
3.466
3.463
3.460
3.457
3.454
3.452
3.449
3.447
3.444
3.442
3.439
3.437
3.435
3.433
3.431
3.429
3.427
3.425
3.423
3.421
3.420
3.418
3.416
3.415
3.413
3.412
3.410
3.409
3.407
3.406
3.405
3.403
3.402
3.401
3.399
3.398
3.397
3.396
3.395
97
98
99
100
110
120
130
140
150
Infinity
1.290
1.290
1.290
1.290
1.289
1.289
1.288
1.288
1.287
1.282
1.661
1.661
1.660
1.660
1.659
1.658
1.657
1.656
1.655
1.645
1.985
1.984
1.984
1.984
1.982
1.980
1.978
1.977
1.976
1.960
2.365
2.365
2.365
2.364
2.361
2.358
2.355
2.353
2.351
2.326
2.627
2.627
2.626
2.626
2.621
2.617
2.614
2.611
2.609
2.576
3.176
3.175
3.175
3.174
3.166
3.160
3.154
3.149
3.145
3.090
3.394
3.393
3.392
3.390
3.381
3.373
3.367
3.361
3.357
3.291
Excel Paired Difference of Means Worksheet
This is used for the situation where the two groups are not independent from e
Our startegy will be to take the difference between the two groups as a single v
The sample size must be the same for each measurement, and the measuresm
so that you are calculating the difference for the same subject in both time peri
You can enter the summary statistics or you can enter the data to the right.
Values in red need to be entered for the problem
Post
n
mean
Std Dev
Variance
20
5.45
2.06
4.244
Pre
20
2.4
1.35
1.823
Difference Ho: Difference
20Confidence Level
3.05
1.76
3.098
Standard Error for test
degrees of freedom
Hypothesis Test
Confidence Interval
t-value
BOE
2.226
0.394
19
Difference
Std Error
3.050
0.394
t* =
p-value
1-tailed
2-tailed
7.750
95
2.093
0.824
to
0.000
0.000
% Level
3.874
0
95
dependent from each other. One example would be the same subjects measured at two time period
oups as a single varaible, and then test to see if the mean difference is zero or not.
nd the measuresment s must be in the same order
t in both time periods.
ta to the right.
Use this if you entered the data to the right
Post
20
18.15
2.87
8.239
n
mean
Std Dev
Variance
Pre
20
7.10
4.05
16.411
Difference
20
11.05
4.02
16.155
Standard Error for test
degrees of freedom
Hypothesis Test
Confidence Interval
t-value
BOE
9.169
0.899
19
Difference
Std Error
11.050
0.899
t* =
p-value
1-tailed
2-tailed
12.295
95
% Level
2.093
1.881
to
0.000
0.000
12.931
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easured at two time periods.
ro or not.
If you have’nt calculated the difference, you can enter or copy the data here for the calculations.
This worksheet will calculate the difference for up to 550 subjects.
N
Mean
Std Dev
Variance
20
18.15
2.87
8.24
Subject
Post
20
7.10
4.05
16.41
Pre
20
11.05
4.02
16.16
Difference
N
Sum Sqrs
Q^2
Q
20
2749
137.45
11.72
Q^2
1
21
7
14
196
2
21
14
7
49
3
20
13
7
49
4
17
3
14
196
5
19
8
11
121
6
14
0
14
196
7
18
5
13
169
8
20
2
18
324
9
19
8
11
121
10
16
9
7
49
11
22
5
17
289
12
16
9
7
49
13
12
1
11
121
14
16
6
10
100
15
22
10
12
144
16
16
11
5
25
17
16
11
5
25
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20
2
18
324
19
20
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24
25
16
22
7
11
9
11
81
121
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for the calculations.
Component Percent
Part due to Var
Part due to Means
Q^2
15.35
122.10
137.45
11.2%
88.8%
Single Mean or Proportion Confidence Interval Worksheet Developed by Dr. Tom Ilvento
Items in red need to be entered by you
MEAN
Std Deviation
Sample Size
Confidence Level
Standard Error
Z or t-value
BOE
Lower
Upper
40.6
1.6
30
95
0.292
PROPORTION
Sample Size
Confidence Level
Std Deviation
Standard Error
Z-value
BOE
Lower
Upper
0.5000
600
95
0.5000
0.0204
1.960
0.0400
0.4600
0.5400
Using Z
1.960
0.573
40.03
41.17
Using t
2.045
0.5974
40.003
41.197
d by Dr. Tom Ilvento
Small Sample Hypothesis Test and Confidence Interval, Sigma Unknown
Developed by Dr. Tom Ilvento
Items in red are values you need to enter
Hypothesis Test using t-distribution
Sample Mean
12648.89
Sample std dev
1898.67
Sample Size
20.00
Null Value
11500.00
Alpha Value
0.050
Confidence Level
95
d.f.
19.00
Std Error
424.556
T.S. t*
2.71
p-value 1-tailed
p-value 2-tailed
0.0070
0.0140
Conclusion
Reject Ho:
Reject Ho:
1-tailed Critical t Value
2-tailed Critical t Value
-1.729
-2.093
1.729
2.093
Confidence Interval using t distribution
MEAN
12648.889
Std Deviation
1898.673
Sample Size
20.000
Confidence Level
0.950
Standard Error
424.556
t-value
2.093
BOE
888.606
Lower 11760.283
Upper 13537.495
-5.0 -4.0
-5.0 -4.0
For a Proportion
Null Value
Alpha level
Sample Size
Sample Proportion
Variance
Standard Deviation
Std Error
0.5
0.05
100
0.56
0.2464
0.4964
0.0500
T.S.
Critical Value
p 1-tailed
p 2-tailed
1.200
0.1151
0.2301
p-Values for Two-Tailed Test
-4.0 -3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
4.0
5.0
p-Values for One-Tailed Upper
p-Value for One-Tailed Lower
-5.0 -4.0 -3.0 -2.0 -1.0 0.0
-4.0 -3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
4.0
5.0
1.0
2.0
3.0
4.0
4.0
5.0
z
-4.000
-3.867
-3.733
-3.600
-3.467
-3.333
-3.200
-3.067
-2.933
-2.800
-2.667
-2.533
-2.400
-2.267
-2.133
-2.000
-1.867
-1.733
-1.600
-1.467
-1.333
-1.200
-1.067
-0.933
-0.800
-0.667
-0.533
-0.400
-0.267
-0.133
0.000
0.133
0.267
0.400
0.533
0.667
0.800
0.933
1.067
1.200
1.333
1.467
1.600
1.733
1.867
2.000
2.133
2.267
f (x )
0.0001338
0.0002261
0.0003753
0.0006119
0.0009801
0.0015423
0.0023841
0.0036204
0.0054011
0.0079155
0.0113960
0.0161179
0.0223945
0.0305672
0.0409873
0.0539910
0.0698671
0.0888185
0.1109208
0.1360825
0.1640101
0.1941861
0.2258628
0.2580778
0.2896916
0.3194480
0.3460539
0.3682701
0.3850069
0.3954118
0.3989423
0.3954118
0.3850069
0.3682701
0.3460539
0.3194480
0.2896916
0.2580778
0.2258628
0.1941861
0.1640101
0.1360825
0.1109208
0.0888185
0.0698671
0.0539910
0.0409873
0.0305672
F(x)
0.0000317
0.0000552
0.0000945
0.0001591
0.0002635
0.0004291
0.0006871
0.0010823
0.0016767
0.0025551
0.0038304
0.0056492
0.0081975
0.0117053
0.0164487
0.02275 …
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