*Tables give the key values of distribution for example you can use a table out when an estimate z distribution 95% of
sample estimates are within 1.96 standard errors, 5% are outside
*If you use s then you do not have an accurate value of standard error if n is small, the df takes this into account. .
Statistical tables page 1 of 4, You will be given these tables in week 11 and final exam
Two sided (tail) t-table for confidence interval and for testing a two sided hypothesis test
This table can be used in many situations including the following situation where you test the following hypothesis population mean using the sample mean
H0: μ = μ0 ( μ0 is what is given in the question you do not know it is true)
H1: μ ≠ μ0 ( you are checking if μ is different to μ0)
Example for when α=0.05
And test stat has a Z distribution
Go to row z(the bottom row) and column
0.05
And find the critical value 1.96
Do not reject H0 region
The z chart has the
same information
P(Z>1.96) = 0.025
reject H0 regions
If the null hypothesis is true the
probability the test statistic is in the shaded
region is α, The critical value is what
comes from the table
P(Z<-1.96) = 0.025
Many books do not use diagrams they use
the international notation
P(Z>1.96) = 0.025 so Z0.025=1.96
P(Z<-1.96) = 0.025 so -Z0.025=-1.96
For a confidence interval (CI) before you
get the sample There is a
(1-α)×100% probability that the parameter
is within “critical value”דstandard error”
of the statistic.
significance level α is the column
df
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
40
50
60
80
99
100
1000
∞(z*)
0.5
0.4
0.3
0.2
0.1
0.05
±1.000
±0.817
±0.765
±0.741
±0.727
±0.718
±0.711
±0.706
±0.703
±0.700
±0.697
±0.696
±0.694
±0.692
±0.691
±0.690
±0.689
±0.688
±0.688
±0.687
±0.686
±0.686
±0.685
±0.685
±0.684
±0.684
±0.684
±0.683
±0.683
±0.683
±0.681
±0.679
±0.679
±0.678
±0.677
±0.677
±0.675
±0.674
±1.376
±1.061
±0.979
±0.941
±0.920
±0.906
±0.896
±0.889
±0.883
±0.879
±0.876
±0.873
±0.870
±0.868
±0.866
±0.865
±0.863
±0.862
±0.861
±0.860
±0.859
±0.858
±0.858
±0.857
±0.856
±0.856
±0.855
±0.855
±0.854
±0.854
±0.851
±0.849
±0.848
±0.846
±0.845
±0.845
±0.842
±0.841
±1.963
±1.386
±1.250
±1.190
±1.156
±1.134
±1.119
±1.108
±1.100
±1.093
±1.088
±1.083
±1.079
±1.076
±1.074
±1.071
±1.069
±1.067
±1.066
±1.064
±1.063
±1.061
±1.060
±1.059
±1.058
±1.058
±1.057
±1.056
±1.055
±1.055
±1.050
±1.047
±1.045
±1.043
±1.042
±1.042
±1.037
±1.036
±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
±1.303
±1.299
±1.296
±1.292
±1.29
±1.290
±1.282
±1.282
±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
±1.684
±1.676
±1.671
±1.664
±1.66
±1.660
±1.646
±1.645
50%CI
60%CI
70%CI
80%CI
90%CI
0.01
0.005
0.002
0.001
±12.706 ±15.895 ±31.821
±4.303 ±4.849 ±6.965
±3.182 ±3.482 ±4.541
±2.776 ±2.999 ±3.747
±2.571 ±2.757 ±3.365
±2.447 ±2.612 ±3.143
±2.365 ±2.517 ±2.998
±2.306 ±2.449 ±2.896
±2.262 ±2.398 ±2.821
±2.228 ±2.359 ±2.764
±2.201 ±2.328 ±2.718
±2.179 ±2.303 ±2.681
±2.160 ±2.282 ±2.650
±2.145 ±2.264 ±2.624
±2.131 ±2.249 ±2.602
±2.120 ±2.235 ±2.583
±2.110 ±2.224 ±2.567
±2.101 ±2.214 ±2.552
±2.093 ±2.205 ±2.539
±2.086 ±2.197 ±2.528
±2.080 ±2.189 ±2.518
±2.074 ±2.183 ±2.508
±2.069 ±2.177 ±2.500
±2.064 ±2.172 ±2.492
±2.060 ±2.167 ±2.485
±2.056 ±2.162 ±2.479
±2.052 ±2.158 ±2.473
±2.048 ±2.154 ±2.467
±2.045 ±2.150 ±2.462
±2.042 ±2.147 ±2.457
±2.021 ±2.123 ±2.423
±2.009 ±2.109 ±2.403
±2.000 ±2.099 ±2.390
±1.990 ±2.088 ±2.374
±1.984 ±2.081 ±2.365
±1.984 ±2.081 ±2.364
±1.962 ±2.056 ±2.330
±1.960 ±2.054 ±2.326
±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
±2.704
±2.678
±2.660
±2.639
±2.626
±2.626
±2.581
±2.576
±127.321
±14.089
±7.453
±5.598
±4.773
±4.317
±4.029
±3.833
±3.690
±3.581
±3.497
±3.428
±3.372
±3.326
±3.286
±3.252
±3.222
±3.197
±3.174
±3.153
±3.135
±3.119
±3.104
±3.091
±3.078
±3.067
±3.057
±3.047
±3.038
±3.030
±2.971
±2.937
±2.915
±2.887
±2.871
±2.871
±2.813
±2.807
±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
±3.307
±3.261
±3.232
±3.195
±3.175
±3.174
±3.098
±3.090
±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
±3.551
±3.496
±3.460
±3.416
±3.391
±3.390
±3.300
±3.291
95%CI
99%CI
99.5%CI
99.8%CI
99.9%CI
1
0.04
96%CI
0.02
98%CI
Statistical tables page 2 of 4, You will be given these tables in week 11 test and final exam
t
One sided t table (upper tail t-table, most books only give you this upper t-table value, if a book says n-1,α go to row n-1, column
α)
If you use s then you do not have an accurate value of standard error if n is small, the df takes this into account
This table can be used in many situations including the following situation where you need to use the following null and alternate hypothesis for
the a single population mean
H0: μ = μ0
( μ0 is what is given in the question you do not know it is true)
H1: μ > μ0 ( you are checking if μ more than μ0)
Tips for using tables t-table, (the main z scores are at the bottom of the t-table)
Example for when α=0.05
And test stat has a Z distribution
Go to row z and column 0.05
And find the critical value 1.645
The critical values are given in the table
below
If the null hypothesis is true the
probability the test statistic is above the
critical value is α
The z chart has the same
information
P(Z>1.645) = 0.05
Most books do not use diagrams they use
the international notation
P(Z>1.645) = 0.05 or Z0.05 = 1.645
significance level α is the column
df
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
40
50
60
80
99
100
1000
∞(z*)
0.25
0.2
0.15
0.1
0.05
0.025
0.02
0.01
0.005
0.0025
0.001
0.0005
1.000
0.817
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.696
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.686
0.686
0.685
0.685
0.684
0.684
0.684
0.683
0.683
0.683
0.681
0.679
0.679
0.678
0.677
0.677
0.675
0.674
1.376
1.061
0.979
0.941
0.920
0.906
0.896
0.889
0.883
0.879
0.876
0.873
0.870
0.868
0.866
0.865
0.863
0.862
0.861
0.860
0.859
0.858
0.858
0.857
0.856
0.856
0.855
0.855
0.854
0.854
0.851
0.849
0.848
0.846
0.845
0.845
0.842
0.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.042
1.037
1.036
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
1.303
1.299
1.296
1.292
1.29
1.290
1.282
1.282
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
1.684
1.676
1.671
1.664
1.66
1.660
1.646
1.645
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
2.021
2.009
2.000
1.990
1.984
1.984
1.962
1.960
15.895
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.081
2.056
2.054
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
2.423
2.403
2.390
2.374
2.365
2.364
2.330
2.326
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
2.704
2.678
2.660
2.639
2.626
2.626
2.581
2.576
127.321
14.089
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.871
2.813
2.807
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
3.307
3.261
3.232
3.195
3.175
3.174
3.098
3.090
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
3.551
3.496
3.460
3.416
3.391
3.390
3.300
3.291
50%CI
60%CI
70%CI
80%CI
90%CI
95%CI
96%CI
98%CI
99%CI
99.5%CI
99.8%CI
99.9%CI
2
Statistical tables page 3 of 4, You will be given these tables in week 11 and final exam
One sided table (lower tail t-table)
This table can be used in many situations including the following situation where you need to use the following null and alternate hypothesis for
the a single population mean
H0: μ = μ0
( μ0 is what is given in the question you do not know it is true)
H1: μ < μ0 ( you are checking if μ less than μ0)
When you have the lower tail case shade in the left to get the reject H0 region
Reject H0 if the test statistic in the rejection region.
If you use s then you do not have an accurate value of standard error if n is small, the df takes this into account (the
main z scores are at the bottom of this t-table)
Example for when α=0.05
And test stat has a Z distribution
Go to row z and column 0.05
And find the critical value -1.645
Do not reject
H0 region
reject H0 region
area is α
The z chart has the
same information
P(Z<-1.645) = 0.05
If the null hypothesis is true the probability
the test statistic is below the critical value is
α,
Most books do not use diagrams they use
the international notation
P(Z<-1.645) = 0.05 or -Z0.05 = -1.645
The critical value is what comes from the
table
significance level α is the column
0.25
0.2
0.15
0.1
0.05
0.025
df
1
2
3
4
5
6
7
8
9
10
11
-1.000
-0.817
-0.765
-0.741
-0.727
-0.718
-0.711
-0.706
-0.703
-0.700
-0.697
-1.376
-1.061
-0.979
-0.941
-0.920
-0.906
-0.896
-0.889
-0.883
-0.879
-0.876
-1.963
-1.386
-1.250
-1.190
-1.156
-1.134
-1.119
-1.108
-1.100
-1.093
-1.088
-3.078
-1.886
-1.638
-1.533
-1.476
-1.440
-1.415
-1.397
-1.383
-1.372
-1.363
-6.314
-2.920
-2.353
-2.132
-2.015
-1.943
-1.895
-1.860
-1.833
-1.812
-1.796
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
∞(z*)
-0.694
-0.692
-0.691
-0.690
-0.689
-0.688
-0.688
-0.687
-0.686
-0.686
-0.685
-0.685
-0.684
-0.684
-0.684
-0.683
-0.683
-0.683
-0.681
-0.679
-0.679
-0.678
-0.677
-0.675
-0.674
-0.870
-0.868
-0.866
-0.865
-0.863
-0.862
-0.861
-0.860
-0.859
-0.858
-0.858
-0.857
-0.856
-0.856
-0.855
-0.855
-0.854
-0.854
-0.851
-0.849
-0.848
-0.846
-0.845
-0.842
-0.841
-1.079
-1.076
-1.074
-1.071
-1.069
-1.067
-1.066
-1.064
-1.063
-1.061
-1.060
-1.059
-1.058
-1.058
-1.057
-1.056
-1.055
-1.055
-1.050
-1.047
-1.045
-1.043
-1.042
-1.037
-1.036
-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
-1.303
-1.299
-1.296
-1.292
-1.290
-1.282
-1.282
50%CI
60%CI
70%CI
80%CI
0.005
0.0025
0.001
0.0005
-12.706 -15.895 -31.821
-4.303 -4.849 -6.965
-3.182 -3.482 -4.541
-2.776 -2.999 -3.747
-2.571 -2.757 -3.365
-2.447 -2.612 -3.143
-2.365 -2.517 -2.998
-2.306 -2.449 -2.896
-2.262 -2.398 -2.821
-2.228 -2.359 -2.764
-2.201 -2.328 -2.718
-63.657
-9.925
-5.841
-4.604
-4.032
-3.707
-3.499
-3.355
-3.250
-3.169
-3.106
-127.321
-14.089
-7.453
-5.598
-4.773
-4.317
-4.029
-3.833
-3.690
-3.581
-3.497
-318.309
-22.327
-10.215
-7.173
-5.893
-5.208
-4.785
-4.501
-4.297
-4.144
-4.025
-636.619
-31.599
-12.924
-8.610
-6.869
-5.959
-5.408
-5.041
-4.781
-4.587
-4.437
-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
-1.684
-1.676
-1.671
-1.664
-1.660
-1.646
-1.645
-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
-2.021
-2.009
-2.000
-1.990
-1.984
-1.962
-1.960
-2.282
-2.264
-2.249
-2.235
-2.224
-2.214
-2.205
-2.197
-2.189
-2.183
-2.177
-2.172
-2.167
-2.162
-2.158
-2.154
-2.150
-2.147
-2.123
-2.109
-2.099
-2.088
-2.081
-2.056
-2.054
-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
-2.423
-2.403
-2.390
-2.374
-2.364
-2.330
-2.326
-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
-2.704
-2.678
-2.660
-2.639
-2.626
-2.581
-2.576
-3.372
-3.326
-3.286
-3.252
-3.222
-3.197
-3.174
-3.153
-3.135
-3.119
-3.104
-3.091
-3.078
-3.067
-3.057
-3.047
-3.038
-3.030
-2.971
-2.937
-2.915
-2.887
-2.871
-2.813
-2.807
-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
-3.307
-3.261
-3.232
-3.195
-3.174
-3.098
-3.090
-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
-3.551
-3.496
-3.460
-3.416
-3.390
-3.300
-3.291
90%CI
95%CI
96%CI
98%CI
99%CI
99.5%CI
99.8%CI
99.9%CI
3
0.02
0.01
Statistical tables page 4of 4, You will be given these tables in all tests and exams
χ2 values
χ2 Table, Use this to test independence if you are given a two way table
(in rarer cases you can also test the goodness of fit)
df degrees of freedom for χ2 curve (This will usually be 1)
P area under the χ2 curve with df degrees of freedom to the right
Unless you are told other wise assume that
the significance level  is 5%
so use the  =0.05 column
0.25
df
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
40
50
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.93
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
0.20
0.15
0.10
0.05
1.64
3.22
4.64
5.59
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
15.58
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
53.16
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.90
19.4
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.29
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
3.84
5.99
7.81
9.49
11.07
12.53
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
39.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
Tail probabilities α
0.025 0.02
0.01
0.005
0.0025
0.001
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
7.88
10.60
12.84
14.86
16.75
13.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.83
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
5.41
7.82
9.84
11.67
13.33
15.03
16.62
18.17
19.63
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
6.63
9.21
11.34
13.23
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
2
2
This is 40, 0.25 =45.62
Because the df is 40
And the shaded area is 0.25
This is 50, 0.001 =86.66
Because the df is 50
And the shaded area in the right tail is 0.001
4