ANSWERS
(1) (a) ME = 0.1622 and the Confidence interval is (2.1978, 2.5222)
(b) The results can’t be generalized because the sample drawn for the study was not random. The sampling was “convenient sampling” as all students were drawn from a particular locality.
(2) (a) t = 3.856, df = 377, p- value = 0.0001
(b) Confidence interval is (0.2254, 0.6946)
Since 0 falls outside the above interval, the results agree with those in the hypothesis test.
(3) (a) SED = 5.154, t = -1.222, df = 23, p- value = 0.234
Here are the complete details of the steps …
Data:
n1 = 20
n2 = 5
x1-bar = 8.2
x2-bar = 14.5
s1 = 10.7
s2 = 8.2
Hypotheses:
Ho: μ1 = μ2
Ha: μ1 ≠ μ2
Decision Rule:
α = 0.05
Degrees of freedom = 20 + 5 - 2 = 23
Lower Critical t- score = -2.068657599
Upper Critical t- score = 2.068657599
Reject Ho if |t| > 2.068657599
Test Statistic:
Pooled SD, s = Ö[{(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)]
= √(((20 - 1) * 10.7^2 + (5 - 1) * 8.2^2)/(20 + 5 -2))
= 10.31
SE = s * Ö{(1 /n1) + (1 /n2)} = 10.3088606885364 * √((1/20) + (1/5)) = 5.154430344
t = (x1-bar -x2-bar)/SE = -1.222249517
p- value = 0.23399174
Decision (in terms of the hypotheses):
Since 1.222249517 < 2.068657599 we fail to reject Ho
Conclusion (in terms of the problem):
There is no sufficient evidence to reject the hypothesis that the two groups consume equal amounts of drinks
(b) The Confidence interval is (-16.9628, 4.3628)
(4)
Df / t* / z*2 / 4.3027 / 1.96
3 / 3.1824 / 1.96
4 / 2.7764 / 1.96
5 / 2.5706 / 1.96
6 / 2.4469 / 1.96
7 / 2.3646 / 1.96
8 / 2.3060 / 1.96
9 / 2.2622 / 1.96
10 / 2.2281 / 1.96
11 / 2.2010 / 1.96
12 / 2.1788 / 1.96
13 / 2.1604 / 1.96
14 / 2.1448 / 1.96
15 / 2.1314 / 1.96
16 / 2.1199 / 1.96
17 / 2.1098 / 1.96
18 / 2.1009 / 1.96
19 / 2.0930 / 1.96
20 / 2.0860 / 1.96
21 / 2.0796 / 1.96
22 / 2.0739 / 1.96
23 / 2.0687 / 1.96
24 / 2.0639 / 1.96
25 / 2.0595 / 1.96
26 / 2.0555 / 1.96
27 / 2.0518 / 1.96
28 / 2.0484 / 1.96
29 / 2.0452 / 1.96
30 / 2.0423 / 1.96
31 / 2.0395 / 1.96
32 / 2.0369 / 1.96
33 / 2.0345 / 1.96
34 / 2.0322 / 1.96
35 / 2.0301 / 1.96
36 / 2.0281 / 1.96
37 / 2.0262 / 1.96
38 / 2.0244 / 1.96
39 / 2.0227 / 1.96
40 / 2.0211 / 1.96
41 / 2.0195 / 1.96
42 / 2.0181 / 1.96
43 / 2.0167 / 1.96
44 / 2.0154 / 1.96
45 / 2.0141 / 1.96
46 / 2.0129 / 1.96
47 / 2.0117 / 1.96
48 / 2.0106 / 1.96
49 / 2.0096 / 1.96
50 / 2.0086 / 1.96
51 / 2.0076 / 1.96
52 / 2.0066 / 1.96
53 / 2.0057 / 1.96
54 / 2.0049 / 1.96
55 / 2.0040 / 1.96
56 / 2.0032 / 1.96
57 / 2.0025 / 1.96
58 / 2.0017 / 1.96
59 / 2.0010 / 1.96
60 / 2.0003 / 1.96
61 / 1.9996 / 1.96
62 / 1.9990 / 1.96
63 / 1.9983 / 1.96
64 / 1.9977 / 1.96
65 / 1.9971 / 1.96
66 / 1.9966 / 1.96
67 / 1.9960 / 1.96
68 / 1.9955 / 1.96
69 / 1.9949 / 1.96
70 / 1.9944 / 1.96
71 / 1.9939 / 1.96
72 / 1.9935 / 1.96
73 / 1.9930 / 1.96
74 / 1.9925 / 1.96
75 / 1.9921 / 1.96
76 / 1.9917 / 1.96
77 / 1.9913 / 1.96
78 / 1.9908 / 1.96
79 / 1.9905 / 1.96
80 / 1.9901 / 1.96
81 / 1.9897 / 1.96
82 / 1.9893 / 1.96
83 / 1.9890 / 1.96
84 / 1.9886 / 1.96
85 / 1.9883 / 1.96
86 / 1.9879 / 1.96
87 / 1.9876 / 1.96
88 / 1.9873 / 1.96
89 / 1.9870 / 1.96
90 / 1.9867 / 1.96
91 / 1.9864 / 1.96
92 / 1.9861 / 1.96
93 / 1.9858 / 1.96
94 / 1.9855 / 1.96
95 / 1.9853 / 1.96
96 / 1.9850 / 1.96
97 / 1.9847 / 1.96
98 / 1.9845 / 1.96
99 / 1.9842 / 1.96
100 / 1.9840 / 1.96
As df increases, the t* decreases. At df = 100, the t* = 1.984 is higher than z* = 1.96