Question

Anemia (low healthy blood cells or hemoglobin) has an important role in exercise performance. However, the direct link between rapid changes of hemoglobin and exercise performance is still unknown. A study investigated 18 patients with a blood disorder (beta-thalassemia). Participants in the study performed an exercise test before and the day after receiving a blood transfusion. Data are given in the table.

ID	Change in HB	Obese	RER > 1.1
1	-1.4	No	No
2	-1.5	No	No
3	-2	No	Yes
4	-2.1	No	No
5	-1.9	Yes	Yes
6	-1.6	Yes	No
7	-1.8	No	Yes
8	-0.8	No	Yes
9	-1	No	No
10	-1.2	No	Yes
11	-0.8	No	No
12	-1.5	Yes	No
13	-1.4	No	Yes
14	-2.6	No	Yes
15	-1.7	No	Yes
16	-2.6	No	Yes
17	-2.7	Yes	No
18	-1.5	Yes	No

1. Researchers are interested if the distribution of values for the change in HB (Hemoglobin) follows the pattern where 1/3 of the time the decrease is less than or equal to 1 (HB ≥ -1), 1/3 of the time the decrease is between 1 and 2 (-2 <HB <-1), and 1/3 the time the decrease is greater than or equal to 2 (HB ≤ -2). From our sample what can you conclude?

a. state the null hypothesis

b. which test statistic would you use and what is the observed value

c. which conclusion would you reach (justify)?

Answer 1

a)

The Null hypothesis is defined as,

Null hypothesis: There is no significant difference between the observed proportion and the expected proportion.i.e.

p1=1/3, p2=1/3, and p3=1/3

b)

The Chi-Square Goodness of fit test is used to test whether the sample is from the same distribution.

The observed values are,

	Observed frequency
HB>=-1	3
-2<HB<-1	10
HB<=-2	5
Total	18

The expected frequencies are,

	Observed frequency
HB>=-1	18/3=6
-2<HB<-1	18/3=6
HB<=-2	18/3=6

The Chi-Square statistic is obtained using the following formula,

The chi-square critical value is obtained from chi-square distribution table for = k - 1 = 3 - 1 = 2

Since,

at a 5% significance level, the null hypothesis is not rejected.

c)

Hence . it can be concluded that the sample follows the given distribution.