Question

A knee surgeon collected data on the post-operative hemoglobin levels (in g/dL) of 102 patients who had total knee replacements. He randomly assigned half of the patients to receive a closed suction drain and half to receive no drain. Of interest is whether postoperative hemoglobin levels, on average, differed between the drain and no drain groups. The data are available in the file knee.csv.

KNEE.CSV:

    hemoglobin    group
1          8.4    drain
2          8.7    drain
3         10.7    drain
4          9.1    drain
5          9.1    drain
6         10.9    drain
7          9.5    drain
8          7.6    drain
9          8.2    drain
10         8.5    drain
11        10.3    drain
12         9.4    drain
13         9.4    drain
14         9.1    drain
15         8.4    drain
16        11.0    drain
17         9.5    drain
18         6.8    drain
19         9.8    drain
20         8.5    drain
21         7.8    drain
22         8.8    drain
23         7.9    drain
24         8.2    drain
25         8.3    drain
26         7.1    drain
27         9.9    drain
28         9.2    drain
29         7.7    drain
30        10.4    drain
31         9.5    drain
32         8.7    drain
33        10.0    drain
34        10.0    drain
35         9.9    drain
36         9.8    drain
37         9.6    drain
38         8.9    drain
39         8.7    drain
40         8.6    drain
41         8.2    drain
42         8.8    drain
43         7.6    drain
44        11.4    drain
45        10.3    drain
46         7.8    drain
47         8.6    drain
48         8.5    drain
49         9.9    drain
50         8.9    drain
51         9.3    drain
52         9.1 no.drain
53         9.1 no.drain
54        10.6 no.drain
55         8.9 no.drain
56        10.8 no.drain
57         7.4 no.drain
58         9.7 no.drain
59         9.2 no.drain
60         9.3 no.drain
61         9.5 no.drain
62         8.5 no.drain
63         8.7 no.drain
64         8.0 no.drain
65         7.9 no.drain
66         9.4 no.drain
67         9.6 no.drain
68         9.2 no.drain
69        10.1 no.drain
70        11.4 no.drain
71         8.6 no.drain
72         6.6 no.drain
73        10.2 no.drain
74         8.3 no.drain
75         8.3 no.drain
76        10.2 no.drain
77         8.8 no.drain
78         7.8 no.drain
79         9.3 no.drain
80         8.9 no.drain
81         9.1 no.drain
82         9.5 no.drain
83         8.7 no.drain
84         9.8 no.drain
85         8.9 no.drain
86         9.5 no.drain
87        10.3 no.drain
88         9.6 no.drain
89         8.7 no.drain
90        10.4 no.drain
91        10.2 no.drain
92         9.7 no.drain
93         9.4 no.drain
94         8.4 no.drain
95        10.6 no.drain
96         8.4 no.drain
97        11.5 no.drain
98        10.8 no.drain
99         8.8 no.drain
100        8.0 no.drain
101        8.3 no.drain
102        9.4 no.drain

a. What are the null and alternative hypotheses?

b. (1 mark) What is the value of the test statistic?

c. (1 mark) What is the approximate p-value?

d. Using a significance level of α = 0.05, state your conclusions in the language of the problem.

e. Describe two key assumptions required for the validity of your hypothesis test and explain why they are reasonable in this setting.

Answer 1

Solution

Let

X = post-operative hemoglobin levels (in g/dL) of patients who had total knee replacements and assigned to receive a closed suction drain

Y = post-operative hemoglobin levels (in g/dL) of patients who had total knee replacements and assigned to no drain

Let mean and standard deviation of X be respectively µ₁ and σ₁ and those of Y be µ₂ andσ₂, where σ₁² = σ₂² = σ², say and σ² is unknown.

a. null and alternative hypotheses

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 ≠ µ2 Answer 1

b. value of the test statistic

Test Statistic:

t = (Xbar - Ybar)/{s√(2/n)} = 1.7630 Answer 2

where

s² = (s₁² + s₂²)/2;

Xbar and Ybar are sample averages and

s₁,s₂ are sample standard deviations based on n observations each on X and Y.

c. approximate p-value = 0.081 Answer 3

d. conclusion

Postoperative hemoglobin levels, on average, do not differ between the drain and no drain groups. Answer 4

e. key assumptions required for the validity of ther hypothesis test and explaination on conformity

Assumption 1: The sample averages follow Normal distribution – quite reasonable since n = 51 is large enough for the Central Limit Theorem to hold good. Answer 5

Assumption 2: the population variances are equal – again quite reasonable since sample standard deviations 1.47 and 0.93 are quite close to each other. Answer 6

Details of calculations and Back-up Theory

n	51
Xbar	8.8529
Ybar	9.2824
s1	1.4725
s2	0.9267
s^2	1.513515
s	1.23025
tcal =	1.76295
α	0.05
p-value	0.080964

Distribution, significance Level α, and p-value:

Under H₀, t ~ t_{2n - 2}. Hence, p-value = P(t_{2n - 2} > | tcal |).

Using Excel Functions, Statistical TDIST the above is found to be as shown in the above table.

Decision:

Since p-value > α, H₀ is accepted implying no significant difference in means of the two groups.

DONE