In: Statistics and Probability
Refer to the accompanying data set and construct a 90% confidence interval estimate of the mean pulse rate of adult females; then do the same for adult males. Compare the results.
Males |
Females |
|||
85 |
71 |
79 |
82 |
|
71 |
66 |
94 |
79 |
|
52 |
75 |
59 |
70 |
|
58 |
74 |
65 |
77 |
|
53 |
56 |
54 |
86 |
|
62 |
66 |
83 |
88 |
|
52 |
58 |
76 |
89 |
|
79 |
79 |
86 |
91 |
|
53 |
70 |
88 |
9 |
|
61 |
65 |
57 |
96 |
|
72 |
61 |
38 |
69 |
|
57 |
98 |
64 |
90 |
|
67 |
55 |
85 |
85 |
|
80 |
63 |
75 |
79 |
|
80 |
56 |
81 |
75 |
|
64 |
56 |
64 |
55 |
|
68 |
65 |
64 |
102 |
|
97 |
73 |
80 |
74 |
|
42 |
81 |
58 |
79 |
|
86 |
60 |
61 |
73 |
Construct a 90% confidence interval of the mean pulse rate for adult females.
___ bpm < mu < ___ bpm
(Round to one decimal place as needed.)
Construct a 90% confidence interval of the mean pulse rate for adult males.
___ bpm < mu < ___ bpm (Round to one decimal place as needed.)
Compare the results.
A. The confidence intervals do not overlap, so it appears that adult females have a significantly higher mean pulse rate than adult males.
B. The confidence intervals overlap, so it appears that adult males have a significantly higher mean pulse rate than adult females.
C. The confidence intervals overlap, so it appears that there is no significant difference in mean pulse rates between adult females and adult males.
D. The confidence intervals do not overlap, so it appears that there is no significant difference in mean pulse rates between adult females and adult males.
The t critical at = 0.10, df = n - 1 = 39 is 1.685
The Confidence Interval is given by ME
ME = t critical * s / sqrt(n)
____________________________________
For Females: = 73.98, s = 17.059, n = 40
ME = 1.685 * 73.98 / sqrt(40) = 4.54
Lower Limit = 67.98 - 4.54 = 69.44 = 69.4 (Rounded to 1 decimal place)
Upper Limit = 67.98 + 4.54 = 78.52 = 78.5 (Rounded to 1 decimal place)
The 90% Confidence interval is 69.4 < < 78.5
____________________________
For Males: = 67.2, s = 12.403, n = 40
ME = 1.685 * 12.403 / sqrt(40) = 3.3
Lower Limit = 67.2 - 3.3 = 63.9
Upper Limit = 67.2 + 3.3 = 70.5
The 90% Confidence interval is 63.9 < < 70.5
____________________________________________
(c) Comparing the results. Option C
The confidence intervals overlap, so it appears there is no significant difference in mean pulse rates between adult females and adult males.
___________________________________________
Calculation for the mean and standard deviation:
Mean = Sum of observation / Total Observations
Standard deviation = SQRT(Variance)
Variance = Sum Of Squares (SS) / n - 1, where SS = SUM(X - Mean)2.
Males | Females | |||||||
# | X | Mean | (x - mean)2 | # | X | Mean | (x - mean)2 | |
1 | 85 | 67.180 | 317.552 | 1 | 79 | 73.980 | 25.200 | |
2 | 71 | 67.180 | 14.592 | 2 | 94 | 73.980 | 400.800 | |
3 | 52 | 67.180 | 230.432 | 3 | 59 | 73.980 | 224.400 | |
4 | 58 | 67.180 | 84.272 | 4 | 65 | 73.980 | 80.640 | |
5 | 53 | 67.180 | 201.072 | 5 | 54 | 73.980 | 399.200 | |
6 | 62 | 67.180 | 26.832 | 6 | 83 | 73.980 | 81.360 | |
7 | 52 | 67.180 | 230.432 | 7 | 76 | 73.980 | 4.080 | |
8 | 79 | 67.180 | 139.712 | 8 | 86 | 73.980 | 144.480 | |
9 | 53 | 67.180 | 201.072 | 9 | 88 | 73.980 | 196.560 | |
10 | 61 | 67.180 | 38.192 | 10 | 57 | 73.980 | 288.320 | |
11 | 72 | 67.180 | 23.232 | 11 | 38 | 73.980 | 1294.560 | |
12 | 57 | 67.180 | 103.632 | 12 | 64 | 73.980 | 99.600 | |
13 | 67 | 67.180 | 0.032 | 13 | 85 | 73.980 | 121.440 | |
14 | 80 | 67.180 | 164.352 | 14 | 75 | 73.980 | 1.040 | |
15 | 80 | 67.180 | 164.352 | 15 | 81 | 73.980 | 49.280 | |
16 | 64 | 67.180 | 10.112 | 16 | 64 | 73.980 | 99.600 | |
17 | 68 | 67.180 | 0.672 | 17 | 64 | 73.980 | 99.600 | |
18 | 97 | 67.180 | 889.232 | 18 | 80 | 73.980 | 36.240 | |
19 | 42 | 67.180 | 634.032 | 19 | 58 | 73.980 | 255.360 | |
20 | 86 | 67.180 | 354.192 | 20 | 61 | 73.980 | 168.480 | |
21 | 71 | 67.180 | 14.592 | 21 | 82 | 73.980 | 64.320 | |
22 | 66 | 67.180 | 1.392 | 22 | 79 | 73.980 | 25.200 | |
23 | 75 | 67.180 | 61.152 | 23 | 70 | 73.980 | 15.840 | |
24 | 74 | 67.180 | 46.512 | 24 | 77 | 73.980 | 9.120 | |
25 | 56 | 67.180 | 124.992 | 25 | 86 | 73.980 | 144.480 | |
26 | 66 | 67.180 | 1.392 | 26 | 88 | 73.980 | 196.560 | |
27 | 58 | 67.180 | 84.272 | 27 | 89 | 73.980 | 225.600 | |
28 | 79 | 67.180 | 139.712 | 28 | 91 | 73.980 | 289.680 | |
29 | 70 | 67.180 | 7.952 | 29 | 9 | 73.980 | 4222.400 | |
30 | 65 | 67.180 | 4.752 | 30 | 96 | 73.980 | 484.880 | |
31 | 61 | 67.180 | 38.192 | 31 | 69 | 73.980 | 24.800 | |
32 | 98 | 67.180 | 949.872 | 32 | 90 | 73.980 | 256.640 | |
33 | 55 | 67.180 | 148.352 | 33 | 85 | 73.980 | 121.440 | |
34 | 63 | 67.180 | 17.472 | 34 | 79 | 73.980 | 25.200 | |
35 | 56 | 67.180 | 124.992 | 35 | 75 | 73.980 | 1.040 | |
36 | 56 | 67.180 | 124.992 | 36 | 55 | 73.980 | 360.240 | |
37 | 65 | 67.180 | 4.752 | 37 | 102 | 73.980 | 785.120 | |
38 | 73 | 67.180 | 33.872 | 38 | 74 | 73.980 | 0.000 | |
39 | 81 | 67.180 | 190.992 | 39 | 79 | 73.980 | 25.200 | |
40 | 60 | 67.180 | 51.552 | 40 | 73 | 73.980 | 0.960 | |
Total | 2687.00 | 5999.776 | Total | 2959.00 | 11348.976 |
Males | Females | |
n | 40 | 40 |
Sum | 2687 | 2959 |
Average | 67.2 | 73.98 |
SS(Sum of squares) | 5999.776 | 11348.976 |
Variance = SS/n-1 | 153.8404 | 290.9994 |
Std Dev=Sqrt(Variance) | 12.403 | 17.059 |