From 5eeb3f8ea11c65552e64ba1c0be83563dc283d58 Mon Sep 17 00:00:00 2001
From: Brad Duthie We can calculate the mean of each sample by calculating the mean of each row.
These 20 means are reported below. The standard deviation of the 20 sample means reported above is 0.0338701.
+## [1,] 0.744 0.693 0.690 0.665 0.714 0.689 0.695 0.742 0.730 0.706
+## [2,] 0.728 0.715 0.663 0.716 0.730 0.719 0.742 0.718 0.686 0.681
+ The standard deviation of the 20 sample means reported above is 0.0246173.
Now suppose that we only had Sample 1 (i.e., the top row of data).
-The standard deviation of Sample 1 is \(s =\) 0.1011544.
+The standard deviation of Sample 1 is \(s =\) 0.0722957.
We can calculate the standard error from these sample values below, \[s = \frac{0.1011544}{\sqrt{10}} = 0.0319878.\] The estimate of the standard error from calculating the standard deviation of the sample means is therefore 0.0338701, and the estimate from just using the standard error formula and data from only Sample 1 is 0.0319878.
+ \[s = \frac{0.0722957}{\sqrt{10}} = 0.0228619.\] The estimate of the standard error from calculating the standard deviation of the sample means is therefore 0.0246173, and the estimate from just using the standard error formula and data from only Sample 1 is 0.0228619.
These are reasonably close, and would be even closer if we had either a larger sample size in each sample (i.e., higher \(N\)) or a larger number of samples.
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12.6 The standard error
Sample_1
-0.58
-0.81
-0.72
-0.59
-0.77
-0.73
-0.79
+0.71
+0.82
+0.75
0.68
-0.58
0.86
+0.76
+0.60
+0.75
+0.73
+0.78
Sample_2
-0.73
-0.70
+0.59
+0.64
+0.75
+0.65
0.61
-0.58
-0.90
-0.84
+0.71
+0.72
+0.77
+0.72
0.77
-0.83
-0.68
-0.70
Sample_3
-0.79
-0.80
-0.85
-0.68
-0.79
-0.66
+0.63
+0.69
+0.70
+0.82
+0.59
+0.67
+0.65
+0.78
0.75
-0.68
-0.74
0.62
Sample_4
-0.77
-0.91
0.78
-0.75
-0.55
-0.46
-0.88
-0.71
+0.81
0.68
-0.82
+0.77
+0.49
+0.68
+0.63
+0.63
+0.60
+0.58
Sample_5
-0.82
-0.72
-0.95
-0.85
-0.66
-0.74
+0.79
+0.67
0.68
-0.59
-0.57
-0.85
+0.63
+0.79
+0.68
+0.78
+0.82
+0.51
+0.79
Sample_6
-0.84
-0.60
0.67
-0.64
-0.82
+0.69
0.68
-0.61
-0.55
-0.89
+0.64
+0.67
+0.88
+0.71
+0.62
0.74
+0.59
Sample_7
-0.69
-0.72
-0.77
-0.55
-0.81
-0.76
-0.70
-0.67
-0.85
-0.71
+0.82
+0.82
+0.66
+0.83
+0.62
+0.45
+0.83
+0.62
+0.51
+0.79
Sample_8
-0.65
-0.95
-0.71
-0.60
-0.63
-0.73
-0.69
-0.75
-0.73
0.82
+0.78
+0.77
+0.82
+0.52
+0.74
+0.78
+0.68
+0.67
+0.84
Sample_9
-0.92
-0.68
-0.71
+0.70
0.86
-0.74
0.80
-0.77
-0.87
+0.61
+0.65
+0.73
0.71
-0.75
+0.95
+0.55
+0.74
Sample_10
-0.65
-0.56
-0.89
-0.74
-0.68
-0.39
-0.67
+0.77
+0.75
0.75
+0.64
+0.77
+0.66
+0.77
+0.66
0.58
-0.73
+0.71
Sample_11
-0.76
-0.69
+0.85
+0.79
+0.60
+0.71
+0.51
+0.77
+0.67
+0.83
0.80
-0.68
-0.76
0.75
-0.64
-0.85
-0.91
-0.73
Sample_12
-0.77
-0.63
-0.75
-0.58
-0.68
-0.69
-0.85
-0.54
-0.63
+0.60
+0.70
+0.60
+0.82
+0.82
+0.59
+0.79
0.71
+0.68
+0.84
Sample_13
-0.82
-0.67
-0.86
-0.74
-0.74
-0.60
-0.75
+0.64
+0.66
+0.53
0.57
+0.76
+0.62
+0.84
+0.62
0.80
-0.61
+0.59
Sample_14
-0.51
-0.61
-0.80
-0.65
-0.77
-0.87
-0.74
-0.84
-0.75
-0.61
+0.81
+0.68
+0.78
+0.62
+0.69
+0.76
+0.73
+0.73
+0.79
+0.57
Sample_15
-0.75
-0.69
-0.94
-0.63
+0.76
+0.87
+0.72
0.70
-0.74
-0.88
-0.71
-0.74
-0.68
+0.77
+0.54
+0.66
+0.79
+0.82
+0.67
Sample_16
-0.55
0.61
-0.57
-0.64
-0.62
-0.80
-0.80
-0.68
-0.58
+0.86
+0.66
+0.49
+0.95
+0.56
+0.85
0.67
+0.73
+0.81
Sample_17
-0.72
-0.79
-0.67
-0.60
-0.57
-0.56
-0.53
-0.91
-0.67
-0.57
+0.75
+0.73
+0.63
+0.85
+0.74
+0.81
+0.77
+0.86
+0.65
+0.63
Sample_18
-0.83
-0.83
+0.68
+0.71
+0.77
+0.73
+0.56
+0.94
+0.69
+0.54
+0.79
0.77
-0.78
-0.61
-0.70
-0.86
-0.70
-0.85
-0.63
-Sample_19
-0.90
-0.73
-0.76
0.66
-0.63
-0.58
-0.57
-0.74
-0.73
+0.52
0.80
-
-
+Sample_20
+0.73
+0.67
0.71
-0.72
-0.74
-0.84
+0.63
+0.77
0.65
0.72
-0.51
-0.71
+
+
Sample_20
+0.56
0.68
-0.72
+0.76
+0.76
+0.75
+0.58
+0.73
+0.70
+0.59
+0.70
-## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
-## [1,] 0.711 0.734 0.736 0.731 0.743 0.704 0.723 0.726 0.781 0.664
-## [2,] 0.757 0.683 0.716 0.715 0.746 0.652 0.659 0.756 0.710 0.700
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@@ -392,7 +394,11 @@
@@ -392,7 +394,11 @@
@@ -392,7 +394,11 @@
@@ -392,7 +394,11 @@
@@ -392,7 +394,11 @@
@@ -392,7 +394,11 @@
19.3 Confidence intervals for dif
In Exercises 19.1 and 19.2, you used different approaches for calculating 95% CIs from the normal and t-distributions.
Now, fill in the table below reporting 95% CIs calculated using each distribution from the 4 sites using any method you prefer.
Site | @@ -629,7 +635,7 @@
---|
Next, do the same, but now calculate 99% CIs instead of 95% CIs.
Week 5 focuses making statistical inferences using confidence intervals (CIs) and and introduces the t-interval.
- - - +Week 6 introduces hypothesis testing, and how to use and interpret statistical tests that test whether or not the mean (or median) of a dataset is significantly different from some specific value, or whether two different groups in a dataset have the same mean (or median).
+Chapter 20 introduces hypothesis testing, and what it means in statistics. This chapter discusses the general idea of hypothesis testing, null and alternative hypotheses, and how to correctly interpret p-values.
+Chapter 21 introduces the t-test and its non-parametric alternatives. These tests include the one sample t-test, the independent samples t-test, the paired sample t-test, the Wilcoxon test, and the Mann-Whitney U test. The chapter also explains the assumptions underlying these different tests.
+Chapter 22 guides you through the week 6 practical. The aim of this practical is to practice using and correctly interpreting the tests that are introduced in Chapter 21.
N | +alpha == 0.1 | +alpha == 0.05 | +alpha == 0.01 | +
---|---|---|---|
5 | +0 | +– | +– | +
6 | +2 | +0 | +– | +
7 | +3 | +2 | +– | +
8 | +5 | +3 | +0 | +
9 | +8 | +5 | +1 | +
10 | +10 | +8 | +3 | +
11 | +13 | +10 | +5 | +
12 | +17 | +13 | +7 | +
13 | +21 | +17 | +9 | +
14 | +25 | +21 | +12 | +
15 | +30 | +25 | +15 | +
16 | +35 | +29 | +19 | +
17 | +41 | +34 | +23 | +
18 | +47 | +40 | +27 | +
19 | +53 | +46 | +32 | +
20 | +60 | +52 | +37 | +
21 | +67 | +58 | +42 | +
22 | +75 | +65 | +48 | +
23 | +83 | +73 | +54 | +
24 | +91 | +81 | +61 | +
25 | +100 | +89 | +68 | +
26 | +110 | +98 | +75 | +
27 | +119 | +107 | +83 | +
28 | +130 | +116 | +91 | +
29 | +140 | +126 | +100 | +
30 | +151 | +137 | +109 | +
31 | +163 | +147 | +118 | +
32 | +175 | +159 | +128 | +
33 | +187 | +170 | +138 | +
34 | +200 | +182 | +148 | +
35 | +213 | +195 | +159 | +
36 | +227 | +208 | +171 | +
37 | +241 | +221 | +182 | +
38 | +256 | +235 | +194 | +
39 | +271 | +249 | +207 | +
40 | +286 | +264 | +220 | +
41 | +302 | +279 | +233 | +
42 | +319 | +294 | +247 | +
43 | +336 | +310 | +261 | +
44 | +353 | +327 | +276 | +
45 | +371 | +343 | +291 | +
46 | +389 | +361 | +307 | +
47 | +407 | +378 | +322 | +
48 | +426 | +396 | +339 | +
49 | +446 | +415 | +355 | +
50 | +466 | +434 | +373 | +
+ | 2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +10 | +11 | +12 | +13 | +14 | +15 | +16 | +17 | +18 | +19 | +20 | +
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | +– | +– | +– | +– | +– | +– | +0 | +0 | +0 | +0 | +1 | +1 | +1 | +1 | +1 | +2 | +2 | +2 | +2 | +
3 | +– | +– | +– | +0 | +1 | +1 | +2 | +2 | +3 | +3 | +4 | +4 | +5 | +5 | +6 | +6 | +7 | +7 | +8 | +
4 | +– | +– | +0 | +1 | +2 | +3 | +4 | +4 | +5 | +6 | +7 | +8 | +9 | +10 | +11 | +11 | +12 | +13 | +14 | +
5 | +– | +0 | +1 | +2 | +3 | +5 | +6 | +7 | +8 | +9 | +11 | +12 | +13 | +14 | +15 | +17 | +18 | +19 | +20 | +
6 | +– | +1 | +2 | +3 | +5 | +6 | +7 | +10 | +11 | +13 | +14 | +16 | +17 | +19 | +21 | +22 | +24 | +25 | +27 | +
7 | +– | +1 | +3 | +5 | +6 | +8 | +10 | +12 | +14 | +16 | +18 | +20 | +22 | +24 | +26 | +28 | +30 | +32 | +34 | +
8 | +0 | +2 | +4 | +6 | +7 | +10 | +13 | +15 | +17 | +19 | +22 | +24 | +26 | +29 | +31 | +34 | +36 | +38 | +41 | +
9 | +0 | +2 | +4 | +7 | +10 | +12 | +15 | +17 | +20 | +23 | +26 | +28 | +31 | +34 | +37 | +39 | +42 | +45 | +48 | +
10 | +0 | +3 | +5 | +8 | +11 | +14 | +17 | +20 | +23 | +26 | +29 | +33 | +36 | +39 | +42 | +45 | +48 | +52 | +55 | +
11 | +0 | +3 | +6 | +9 | +13 | +16 | +19 | +23 | +26 | +30 | +33 | +37 | +40 | +44 | +47 | +51 | +55 | +58 | +62 | +
12 | +1 | +4 | +7 | +11 | +14 | +18 | +22 | +26 | +29 | +33 | +37 | +41 | +45 | +49 | +53 | +57 | +61 | +65 | +69 | +
13 | +1 | +4 | +8 | +12 | +16 | +20 | +24 | +28 | +33 | +37 | +41 | +45 | +50 | +54 | +59 | +63 | +67 | +72 | +76 | +
14 | +1 | +5 | +9 | +13 | +17 | +22 | +26 | +31 | +36 | +40 | +45 | +50 | +55 | +59 | +64 | +67 | +74 | +78 | +83 | +
15 | +1 | +5 | +10 | +14 | +19 | +24 | +29 | +34 | +39 | +44 | +49 | +54 | +59 | +64 | +70 | +75 | +80 | +85 | +90 | +
16 | +1 | +6 | +11 | +15 | +21 | +26 | +31 | +37 | +42 | +47 | +53 | +59 | +64 | +70 | +75 | +81 | +86 | +92 | +98 | +
17 | +2 | +6 | +11 | +17 | +22 | +28 | +34 | +39 | +45 | +51 | +57 | +63 | +67 | +75 | +81 | +87 | +93 | +99 | +105 | +
18 | +2 | +7 | +12 | +18 | +24 | +30 | +36 | +42 | +48 | +55 | +61 | +67 | +74 | +80 | +86 | +93 | +99 | +106 | +112 | +
19 | +2 | +7 | +13 | +19 | +25 | +32 | +38 | +45 | +52 | +58 | +65 | +72 | +78 | +85 | +92 | +99 | +106 | +113 | +119 | +
20 | +2 | +8 | +13 | +20 | +27 | +34 | +41 | +48 | +55 | +62 | +69 | +76 | +83 | +90 | +98 | +105 | +112 | +119 | +127 | +
21 | +3 | +8 | +15 | +22 | +29 | +36 | +43 | +50 | +58 | +65 | +73 | +80 | +88 | +96 | +103 | +111 | +119 | +126 | +134 | +
22 | +3 | +9 | +16 | +23 | +30 | +38 | +45 | +53 | +61 | +69 | +77 | +85 | +93 | +101 | +109 | +117 | +125 | +133 | +141 | +
23 | +3 | +9 | +17 | +24 | +32 | +40 | +48 | +56 | +64 | +73 | +81 | +89 | +98 | +106 | +115 | +123 | +132 | +140 | +149 | +
24 | +3 | +10 | +17 | +25 | +33 | +42 | +50 | +59 | +67 | +76 | +85 | +94 | +102 | +111 | +120 | +129 | +138 | +147 | +156 | +
25 | +3 | +10 | +18 | +27 | +35 | +44 | +53 | +62 | +71 | +80 | +89 | +98 | +107 | +117 | +126 | +135 | +145 | +154 | +163 | +
26 | +4 | +11 | +19 | +28 | +37 | +46 | +55 | +64 | +74 | +83 | +93 | +102 | +112 | +122 | +132 | +141 | +151 | +161 | +171 | +
27 | +4 | +11 | +20 | +29 | +38 | +48 | +57 | +67 | +77 | +87 | +97 | +107 | +117 | +127 | +137 | +147 | +158 | +168 | +178 | +
28 | +4 | +12 | +21 | +30 | +40 | +50 | +60 | +70 | +80 | +90 | +101 | +111 | +122 | +132 | +143 | +154 | +164 | +175 | +186 | +
29 | +4 | +13 | +22 | +32 | +42 | +52 | +62 | +73 | +83 | +94 | +105 | +116 | +127 | +138 | +149 | +160 | +171 | +182 | +193 | +
30 | +5 | +13 | +23 | +33 | +43 | +54 | +65 | +76 | +87 | +98 | +109 | +120 | +131 | +143 | +154 | +166 | +177 | +189 | +200 | +
31 | +5 | +14 | +24 | +34 | +45 | +56 | +67 | +78 | +90 | +101 | +113 | +125 | +136 | +148 | +160 | +172 | +184 | +196 | +208 | +
32 | +5 | +14 | +24 | +35 | +46 | +58 | +69 | +81 | +93 | +105 | +117 | +129 | +141 | +153 | +166 | +178 | +190 | +203 | +215 | +
33 | +5 | +15 | +25 | +37 | +48 | +60 | +72 | +84 | +96 | +108 | +121 | +133 | +146 | +159 | +171 | +184 | +197 | +210 | +222 | +
34 | +5 | +15 | +26 | +38 | +50 | +62 | +74 | +87 | +99 | +112 | +125 | +138 | +151 | +164 | +177 | +190 | +203 | +217 | +230 | +
35 | +6 | +16 | +27 | +39 | +51 | +64 | +77 | +89 | +103 | +116 | +129 | +142 | +156 | +169 | +183 | +196 | +210 | +224 | +237 | +
36 | +6 | +16 | +28 | +40 | +53 | +66 | +79 | +92 | +106 | +119 | +133 | +147 | +161 | +174 | +188 | +202 | +216 | +231 | +245 | +
37 | +6 | +17 | +29 | +41 | +55 | +68 | +81 | +95 | +109 | +123 | +137 | +151 | +165 | +180 | +194 | +209 | +223 | +238 | +252 | +
38 | +6 | +17 | +30 | +43 | +56 | +70 | +84 | +98 | +112 | +127 | +141 | +156 | +170 | +185 | +200 | +215 | +230 | +245 | +259 | +
39 | +7 | +18 | +31 | +44 | +58 | +72 | +86 | +101 | +115 | +130 | +145 | +160 | +175 | +190 | +206 | +221 | +236 | +252 | +267 | +
40 | +7 | +18 | +31 | +45 | +59 | +74 | +89 | +103 | +119 | +134 | +149 | +165 | +180 | +196 | +211 | +227 | +243 | +258 | +274 | +
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