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66
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71
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4
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69
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3
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4
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70
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62
68
62
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63
67
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62
74
73
74
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3.4
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3.3
4
3.6
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4
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70
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64
66
73
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71
71
63
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70
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3.7
3.7
3.6
3.8
4
3.7
3.6
3.5
3.95
3.6
3.83
3
3.7
3.85
3.33
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4
3.1
3.98
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3.4
3.76
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3.5
3.5
3.6
3.3
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66
80
60
60
80
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84
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80
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72
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72
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69
68
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66
66
69
72
72
72
70
74
76
66
65
70
64
62
60
66
66
73.5
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62
60
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3.7
3.9
3.9
3.9
4
4
3
3.8
3.5
3.7
3.18
3.9
3
3.8
3.75
3.3
4.42
3.5
3.4
3
3.82
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3.5
3.39
3.75
3.6
3
3.78
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4
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3.9
3.6
3.7
3.4
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70
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100
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80
90
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80
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72
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67
72
70
72
71
66
71
73
68
75
72
71
68.5
74
68
73
64
70
73
72
70
70
64
72
62
68
64
64
68.5
67
74
76
73
65
73
66
60
63
67
70
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3.53
3.4
3.57
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3.67
3.98
3.8
3.02
3.7
3
3.79
3.5
2.6
3.8
3.9
3.85
2.85
3.7
3.9
3.94
3.33
3.9
3.74
3.4
3
3.54
3.7
4
3.7
3.8
3.5
3.6
3.87
3.68
2.8
3.5
3.46
3.38
3.77
3.85
3
68
60
68
60
84
56
100
64
76
76
80
84
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64
70
62
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56
68
77
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60
80
60
64
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54
60
60
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64
7
75
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65
63
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84
80
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74
73
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68
65
68
73
67
68
66.5
71
73
67
65
66
76
66
64
74
62
64
70
70
68
72
63
64
71
70
77
71
66
65
61
63
66
72
69
73
74
69.75
73
74
64
3
3.6
4
3.44
3.7
3.6
3.8
3.38
3.4
3.9
4
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3.9
3.8
3.6
3.2
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3.89
3.3
3.04
3.5
3.2
3.9
3.9
3.39
3.97
3.8
3.8
3.8
3.8
3.9
3.5
4
3.5
3.2
3.71
2.7
3.8
3.6
4
3.69
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66
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114
33
66
50
76
90
60
80
90
60
80
70
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100
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100
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70
80
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60
60
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66
62
65
74
69
67
73
66
74
67
69
76
75
70
65
65
68
62
67
71
74
64
62
67
68
72
68
73
70
69
69
69
66
69
63
65
62
75
73
3.9
3.42
3.6
3.7
2.5
3.6
3.6
3.4
2.85
3.89
3.7
3.2
3.8
3.89
4
4
3.7
3.5
3.8
3.7
3.85
3.8
3.75
3.9
3.945
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3.8
3.3
3.7
3.9
2.4
3.93
3.59
3.5
3.79
3.7
3.3
3.7
3.86
3.57
3.85
4
3.3
3.7
68
65
70
90
100
80
70
100
90
60
100
100
90
120
90
95
60
90
110
54
60
55
62
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70
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60
101
60
80
50
70
80
60
60
90
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70
60
80
70
70
75
74
68
65
67
62
67
67
72
66
70
68
75
69.5
68
72
74
68
75
71
72
73
64
74
70
71
64
64
68
59
65
66
70
72
75
67
74
67
71
62
70
72
62
71
64.5
3
3
3.65
3.6
3.9
2.8
3.9
3.2
3.3
3.4
3.79
2.8
3.55
3.4
3.87
3.2
3.5
3.4
3.2
3.8
2.3
3.9
3.65
3.83
3.94
3.85
3.25
3.4
3.5
3.7
3
3.5
3.3
3.2
3.5
3.4
3.4
3.31
3.6
3.7
4.2
3.9
3
3.78
80
52
100
70
80
60
120
130
90
70
72
56
48
85
64
76
72
84
84
68
45
95
60
68
64
72
68
88
80
80
76
70
60
50
70
66
12
70
78
70
72
72
60
120
70
66
72
66
73
63
64
66
70
68
74
68
63
68
63
70
71
65
67
76
71
65.5
64
64
69
65
72
70
66
56
67
68
67
72
63
63
69
64.5
72
71.5
66
61
61
64
3.5
3.8
3.9
3.89
3.3
3.9
3.9
3.65
3.4
4
3.3
3.82
4
3.65
4.032
3.4
3.2
4
3.8
3.73
3.4
3.89
3.8
3.4
3.5
3.89
3.3
3.97
3.87
4.2
3.7
4
4
3.5
3.6
3.5
3.98
3.7
3.5
3.9
3.6
3.4
3.9
3.95
90
80
64
56
80
84
71
72
84
70
65
53
78
65
54
60
75
67
70
74
70
62.5
52
115
52
70
62
70
70
50
120
85
75
70
60
92
69
80
84
72
68
80
80
80
76
60
69
71
69
63
68
70
64
61
64
65
66
69
69
65
67
75
68
72
70
70
72
73
66
68
67
63
70
72
74
69
62
75
70
70
74
72
72
69
72
60
68
75
3.1
3.3
3.4
3.3
3.43
4
3.8
3.85
3.84
3.9
3.78
3.6
3.5
4
3.5
3.6
3.87
3.8
3.4
3.54
3.5
3.94
3.25
3
3.57
3.6
3.4
3.7
3.48
3.72
3.78
3.67
3.991
3.2
3.89
4.21
3.3
3.8
4
3.65
3.41
3.09
3.6
3.97
60
96
88
60
80
80
52
88
100
88
60
76
84
88
72
88
80
80
72
75
88
64
96
80
96
132
64
73
80
76
80
80
72
64
84
84
80
58
110
55
90
77
33
65
67
68
70
76
70
68
63
66
70
69
68
66.5
68
69.5
64
66
72
71
64
68
70
73
66
62
66
70
63
60
62
64
65
72
66
65
66
68
69
75
64
74
60
67
67
62.5
3.8
3.86
2.7
3.9
3.3
3.8
3.75
3.4
3.9
3.64
3.98
3.5
3.7
3.8
3.8
3.8
3.8
3.6
4
3.4
3.97
4
3.94
3.97
4
4
3.7
3.93
3.92
3.97
4.082
4
4
4
3.8
3.89
3.7
3.875
3.75
3.7
3.9
3.6
3.8
3.8
76
84
90
80
50
100
90
60
80
80
80
100
60
110
120
60
70
60
111
60
100
100
90
80
160
80
100
90
90
120
90
60
90
90
100
100
70
80
100
90
90
100
100
120
69
68
72
64
71
67
68
69
64
71
66
72.5
76
68
67
68
70
63
64
72
67
71
72
72
73
66
63
72
70
61
68
70
68
60
63
65
58.5
68
66
60
64
68
68
74
4
3.8
3.88
3.54
3.7
3.5
3.79
3.15
3.98
3.8
3
3.67
3.8
4.2
3.5
3.34
3.7
3.4
3.5
3.5
3.4
4
2.9
3.7
2.8
3.9
3.15
3.2
3
3.67
3.8
3.7
3.3
3.9
3.9
3.7
3.6
3.2
3.5
3.6
3.3
3
3.6
3.5
90
100
80
90
50
70
90
70
90
100
80
60
60
65
80
80
80
80
70
70
80
100
54
72
72
170
42
72
54
100
90
78
66
42
72
100
72
90
78
66
84
63
80
60
72
67
64
68
72
71
67.5
64
70
64
66
71
68
72
74
71
74
65
72
72
63
67
68
70
68
67
66
70
67
73
67
70
69
71
70
70
66
62
70
67
61
68
75
68
3.39
3.2
3.8
3.4
3.7
2.8
4
3.8
3.885
3.93
3.8
3
4
3.2
3.3
3.5
3.8
3.7
2.8
3.3
3.942
3.9
3.25
3.5
3.4
4
4
3.8
3.725
3.9
3.7
3.98
3.75
3.25
3.5
3.8
3.98
3.6
3.6
3.98
3.65
3.96
3.4
3.92
75
60
110
44
60
50
55
80
90
50
85
68
62
56
70
67
59
55
80
62
60
58
160
80
64
57
0
120
80
55
0
80
79
80
90
68
65
54
110
55
68
60
50
62
67
65
73
77
65
73
68
75
73
67
67
66
65
68
68
56
70
70
72
71
67
70
72
72
68
67
68
67
68
65
64
64
73
60
73
72
70
65
65
65
73
62
66
70.5
3.89
3.97
3.2
3.75
3.85
3.5
3.7
3.87
2.7
3.7
3.8
3.9
3.7
3.63
3.8
3.4
3.5
3.6
3
3.2
3.65
3.75
2.75
4
3.1614
3.2
3.6
4.5
2.65
4
3.8
2.5
3.5
3.5
3
3.5
3.86
4
3.67
3.5
3.9
3.93
3.9
2.9
76
80
80
64
60
60
60
80
52
88
88
70
80
78
76
60
72
64
76
73
60
64
96
52
80
72
68
56
76
68
76
96
84
72
65
48
80
60
60
76
68
64
80
64
67
74
71
67
67
61
60.5
61
67
64
64
72
65
68
74
70
78
72
68
64
63
64
65.5
71
64
72
64
71
73
64
72
72
69
70
73
70
68
72
72
70
70
66
68
65
3
2.6
3.3
3.6
3.78
3.8
3.7
3.94
3.98
3.7
3.68
3.88
3.72
3.5
3.6
3.99
3.83
3.33
3.6
3.5
3.7
3.7
3.9
3.6
3.92
3.7
3.25
3.5
3.33
3.8
3.7
3.4
4
3
4
4
3.2
3.4
3.6
3.49
3.87
4
3.7
3.95
30
56
60
64
64
44
44
88
92
96
64
70
80
70
105
74
80
70
68
80
64
84
64
80
64
72
68
68
80
84
72
72
60
52
60
68
76
56
55
120
42
55
13
60
75
65
76
67
72
72
71
75
67
72.5
72
65
69
75
74
62
64
66
76
71
72
67
67
64
67
71
74
64
75
75
70
68
73
65
70
63
69
65
63
63
73
68
65
64
3.3
3.8
3.71
3.83
3.3
2.7
3.8
3.74
3.67
3.86
3.4
3.65
3.9
3.5
3.2
3.8
3.4
3.53
3.6
3.75
3
3.8
3.98
3.8
3.3
4
3.2
3.967
3.5
3.1
3.1
3.89
3.93
3.9
3.6
4
3.5
3.89
3.6
3.98
3
3.87
4
3.65
90
75
60
88
62
87
64
70
25
72
60
72
66
66
84
54
60
90
66
60
60
54
72
60
60
48
60
74
66
60
78
68
68
65
66
88
75
83
84
80
84
63
65
64
75
62
64
74
66
71
67
74
62
73
69
69
71
70
70
67
61
70
67
69
65
66
73
62
64
66
62
66
66
68
70
70
74
63
80
68
64
64
69
62
71
72
68
73
3
3.9
3.2
3
3.6
3.2
3
3.5
4
3.32
4
3.4
3.8
3.75
3.85
3.95
3.2
3.78
3.87
3.2
3.7
3
3.73
3.65
3.94
3.1
3.75
4.02
3.975
3.7
3.86
3
3.8
3.8
3.1
3.96
3.6
3.8
3
3.9
4
3.1
3.4
3.2
64
78
70
82
78
70
80
65
88
140
76
60
64
120
67
54
63
58
52
70
63
50
84
52
34
65
85
78
70
130
88
60
75
65
60
72
35
80
70
75
70
70
80
80
64.5
63
63
62
66
73
73
72
67
65
61
65
77
71.5
61.5
69
65
66
70
71
75
72
76.5
75
72
70
69
71
68
67
68
69
75
66
65
72
67
63
66
62
72
69
63
70
3.46
3.8
3.93
3.2
3.7
3.3
3.1
3.45
3.4
4
3.85
4
3.17
3
4
3.1
3.98
3.5
3.35
3.9
4.1
3.8
3.3
3.52
4
3.65
3.29
3.98
3.2
3.94
3.84
3.65
3.2
3.87
3.4
3.5
3.7
3
3.7
3.5
3.3
4
4
4
120
90
80
70
65
100
60
70
100
70
90
70
60
80
76
60
90
45
88
88
80
60
48
73
68
40
80
77
84
90
62
90
44
62
63
50
60
90
80
110
70
110
50
70
66.5
65
65
62.5
70
64
62
76
66
67.5
69
70
70
73
73
68
65
74
68.5
63
57
62
66
73
76
72
71
68
66
66
75
69
66
66
67
70
66
65
67
66.5
74
67
64
71
4
3.9
3.98
4
3.5
3.89
3.8
3.87
3.95
3.31
3.5
3.6
3.3
3.01
3.8
3.87
4
3.3
3.6
3.8
3.9
3.8
3.3
3
3.4
4
3.4
4
3.97
3.9
3.35
3.8
3.75
3.9
3.5
3.7
3.8
4.6
3.2
3.9
3.9
3.7
3.85
3.3
70
70
100
90
60
70
70
80
70
100
80
88
60
80
64
72
80
68
96
72
56
68
56
56
70
56
70
72
88
44
88
94
72
84
48
60
90
108
84
84
66
78
102
90
65
67.5
63
66
60
67
67
68
65
65
74
65
55
64
65
67
70
71
67.5
70
73
70
62
69
63
70
74
72
70
68.5
68
74.5
71
67
66
71
74
73
65
47
65
73
74
70
3.81
3.2
3.83
3.88
3.43
3.9
3.75
3.54
3
3.6
3.3
3.5
3.8
3.55
3.5
3.76
3.2
3.3
3.6
3.82
3.7
3.4
3.78
3.3
4
3.1
3.8
3.4
3.7
3.95
3.5
3.2
3.4
3.9
3.85
3.74
3.3
3.3
3.86
3.85
3.8
3.6
3.6
3.97
96
96
72
84
96
102
78
96
108
100
72
29
85
83
80
62
27
75
90
60
60
40
45
48
76
60
72
68
56
84
60
72
68
96
64
76
76
92
60
120
104
80
56
130
68
72
59.5
83
69
74
62
66
66
68
67
73
65
62.5
64
65
68
72
71
77
61
64
65
61
63
65
64
64
59
65
69
69
65
64
71
67
74
77
70
73
74
70
67
62
3.84
3.67
4.069
3.8
3.3
3.4
3.8
3.9
3.6
3.8
3.8
3.2
4
3.8
3.84
3.96
3.3
2.98
3.3
3.7
3.3
4.03
3.2
3.9
3.9
3.8
3.99
4.1
3.89
4
3.9
3.98
3.76
4
3.97
3.3
3.9
3.4
3.9
3.5
3.8
3.95
3.79
3.97
80
68
10
76
68
72
60
56
64
80
88
68
94
96
80
52
80
84
56
76
84
80
56
92
80
44
70
80
50
70
70
100
70
100
54
80
60
70
65
76
90
60
80
70
70
72
71
64
66
69
71
62
70
72
72
69
70.75
68
67
70
64
70
62
68
66
72
69
73
63
63
69
67
60
70.5
64
70
66
73
63
73
66
60
63
68
66
64
63
63
3.95
4
3.9
3.7
3.75
3.76
3
3.4
3.72
3.6
2.99
3
3
3.5
3.95
3.3
3.83
3.2
3.6
3.6
3.78
3.4
3.46
3.3
4.04
3.6
3.94
3.62
3.9
3.67
3.5
3.8
4
3
3.8
3.53
4
4.2
3.8
3.5
3.2
3.87
4
4
60
70
50
60
90
90
70
80
70
55
70
80
60
80
90
70
80
85
100
70
70
84
108
66
63
63
72
66
72
96
60
72
72
96
72
66
66
56
90
60
48
48
60
60
71
67
69
73
72
70
72
74.4
66
62
62
65
68
72
67
65
63
65
63.75
67
66
65
67
72
70
74
71
65
69
65
76
70
63
66
70
64
63
66
62
61
68
60
68
66
3.87
3.1
3.5
3.87
3.8
3.9
3.7
3.22
3.8
4
4
3.83
3.7
3.47
3.2
3.2
3.8
3.7
3.75
3.6
3.61
3.5
2.9
3.2
2.4
3.6
3.5
3.3
3.9
4.2
3.7
3.2
3.3
3.2
3.5
3.4
3.6
3.99
3.6
3.8
3.49
3.7
3.1
3.67
72
90
60
66
90
66
84
72
66
102
126
102
72
72
82
72
96
102
60
60
72
76
68
80
72
72
100
90
70
90
67
78
46
72
72
84
60
72
78
72
84
72
84
72
67
70.5
66
69
73
64
66
74.5
67
66
69
70
73
66
75
72
72
65
72
65
74
66
67
65
65
66
69
71.5
73
67
65
63
73.5
70
77
71
77
74
74
62
71
67
66
63
3.7
3.2
3.5
3.98
3.5
3.6
4
3.4
3.9
3.8
4.3
3.7
3.65
3.3
3.51
3.56
3.3
4
3.25
4.2
3
4
3.5
3.7
4
3.5
3.9
3.5
3.4
3.8
3.4
3.86
3.6
3.875
3.77
3.5
3.4
3
3.7
3.3
3.4
4
3.56
3.7
78
78
108
78
92
72
60
90
110
60
58
66
72
70
90
145
80
76
82
86
60
68
88
47
82
60
42
70
60
90
60
70
80
80
70
80
60
70
66
70
90
80
80
100
58
65
65
74
68
66
69
63
64
75
60
68
66
62.5
69
62
66
67
65
71
67
68
62
72
66
72
76
70
61
66
73
67
74
69.5
64
62
71
70
62
72
70
62
69
72
3.9
3.75
3.86
4
3.3
3.25
3.9
3.86
3.5
3.8
4.125
3.9
3.4
3.7
3.2
3
3.3
3.6
3.8
3.72
4
4.7
3.5
3.65
3.8
3.56
3.3
3.2
3
3.48
4
3.6
3
3.3
3.4
3.5
3.6
3.95
3.89
3.5
3.84
3.6
3.6
3.9
60
60
80
88
55
78
68
64
76
72
66
66
66
60
74
60
76
66
67
70
90
80
60
120
90
80
40
90
80
80
70
120
80
60
60
70
60
55
47
68
80
68
80
56
62.5
69
75
72
64
67
69
67
63
66
73
74
59
75
71
69
73
65
67
80
65
72
68
65
70
67
71
69
68
68
67
58
66
64
65.5
66
64
60
62
69
67
68
72
63
3.8
3.9
3.3
3.85
3.8
3.7
4
3.5
3.6
4
4
3
3.8
3.95
3.6
4
3.87
3.86
3
3.8
3.3
3
3.75
3.4
3.5
3.97
3
4
3.91
3.6
3.23
3.98
3.33
3.8
3.3
3.6
3.8
3.88
3.9
3.2
4
3.6
3.96
4.208
48
56
76
68
76
64
72
68
80
80
56
68
80
48
80
96
52
76
80
76
80
80
76
100
64
88
80
76
72
80
40
68
92
48
70
80
65
64
48
51
84
74
63
59
68
63
72
68
72
69
65
66
70
64
66
62
60
63
72
68
65
69
70
63
75
67
72
66
75
66
67
63
64
71
71
72
69
71
71
69
66
76
74
72
74
67
72
63
3.65
3.98
3
3.95
3.5
3.8
3.3
3.75
3.6
3.7
3.5
4
3.8
3.2
3.5
3.6
3.98
3.5
3.87
3.71
4
3.2
3
3.4
3.75
3.82
3.3
4
3.9
3.89
3.6
3.7
3.86
4
4
3.5
4.2
3.85
2.65
3.9
3.68
3.6
3.95
3.7
70
65
100
68
76
60
80
82
55
60
66
60
64
82
44
78
52
68
100
75
66
62
60
64
80
90
7
66
60
56
90
70
90
80
70
7
60
60
70
100
80
60
60
80
68
74
61
71
68
64
62
73
69
70
66
67
64
68
71
68
64
72
74
65
71
74
72
64
70
68
66
70
64
68
63.5
65
66
68
66
70
68
63
75
68.5
57
69
65
72
3.75
3.1
3.89
3.6
3.86
3.9
3.8
3.05
3.75
3.58
3.8
3.15
3.46
3.7
3.66
3.24
3.85
3.5
3.3
3.3
3.4
3.6
3.05
3.4
3.7
3.4
3.1
3
4
4
3.45
3.6
4
3.2
4
3.5
3.8
3.8
3.2
4
4
4.15
3.8
3.96
50
60
50
80
60
80
70
80
60
67
2.3
56
74
96
64
64
60
64
120
120
80
80
68
80
50
64
90
72
72
51
76
72
79
82
80
76
10
56
66
72
65
90
58
70
65
66
77
70.5
73
74
72
68
66
4
3.83
2.9
4.1
3.9
3
3.6
3.65
3.7
62
62
70
58
70
78
80
64
110
STA 100 SSI 2019
R Homework 2 – Due Friday, July 12th by 5pm.
Book homework is for practice.
1
Book Homework
1. For a standard normal random variable (i.e, for a Z ∼ N (µ = 0, σ = 1) find the following:
(a) P (Z < 1.2) (b) P (Z > −2.3)
(c) P (−1 < Z < 0.31) (d) The 97.5th percentile. 2. Assume that heart rate (in beats per minute, or bpm) for STA 100 students is distributed normal, with a mean of 110 bpm and a standard deviation of 20.2 bpm. Assume all students in the following problem are selected from this population. (a) Find the probability that a randomly selected students heart rate is above 125. (b) What percentage of randomly selected students could we expect to have a heart rate between 90 and 130? (c) What is the first and third quartile of heartrates for randomly selected students? (d) What is the 8th percentile for heart rates among randomly selected students? 3. The heights of men in a certain population follow a normal distribution with mean 69.7 inches and standard deviation 2.8 inches. (a) If 10 men are selected at random, what is the probability their average height is above 72 inches? (b) If exactly one man is selected at random, what is the probability his average height is above 72 inches? (c) What is the 80th percentile for the average height of 10 men? (d) What is the probability the average height of 10 men is between 70 and 71 inches? 4. Over 9 months, a random sample of 100 women were asked to record their average menstrual cycle length (in days). The sample average was 28.86 days, with a sample standard deviation of 4.24 days. (a) Calculate the 90% confidence interval for the true average menstrual cycle length. (b) Interpret the confidence interval found in (a) in terms of the problem. (c) A researcher hypothesized that womens menstrual cycles are typically the same length as a lunar month – 29.5
days. Does your interval from (a) support this hypothesis? Explain.
(d) List two hypothetical ways we could reduce the width of this confidence interval.
5. Continue with the data from Problem 4.
(a) If we wanted our confidence interval to have a margin of error of 0.5 days at 90% confidence, how many women
should we sample (at least)?
(b) If we wanted our confidence interval to have a margin of error of 0.1 days at 90% confidence, how many women
should we sample (at least)?
(c) What tends to happen to the sample size we need as the margin of error decreases? Explain based on (a) to (b),
or in plain English.
(d) What tends to happen to the sample size we need as the standard deviation increases? Explain.
1
6. Two treatment costs for a particular medical issue are being compared. The average cost of 210 randomly selected
subjects for treatment A was $490, with a standard deviation of $32. The average cost for 180 randomly selected
subjects for treatment B was $500, with standard deviation $48. Assume the d.f = 300.
(a) Find the 99% confidence interval for the difference costs.
(b) Interpret your interval from (a) in terms of the problem.
(c) Does your interval suggest that the costs are different? Explain.
(d) If we increased the confidence level (i.e, (1 − α)100%), would the new interval be wider or narrower than that found
in (a)? Explain (you do not need to calculate the interval).
7. Two baby-food manufacturers are competing, and manufacturer of brand A believes that their brand causes more
weight gain in infants than manufacturer of brand B. A sample of size 15 fed their babies brand A for two months, and
saw an average weight gain of 36.93 ounces, with a standard deviation of 4.23 ounces. A sample of size 25 fed their
babies brand B and saw an average weight gain of 31.36 ounces with a standard deviation of 3.35 ounces. Assume the
d.f. = 24.
(a) Create a 95% confidence interval for the difference in weight gain between the two baby food brands.
(b) Interpret your interval from (a) in terms of the problem.
(c) Does your interval suggest one brand of baby food …
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