Question

A study is designed to investigate whether there is a difference in response to various treatments in patients with rheumatoid arthritis. The outcome is patient’s self-reported effect of treatment. The data are shown below. Are symptoms independent of treatment? Conduct a ChiSquare test at a 5% level of significance. 1.df= (2 points) 2.Critical value: (2 points) 3.Computed statistic: (2 points) 4.Based on comparing the computed statistics to the critical value which of the following is (are) true? (4 points) a. There is significant evidence, alpha=0.05, to show that treatment and response are not independent. b. There is not significant evidence, alpha=0.05, to show that treatment and response are not independent. c. There is significant evidence, alpha=0.05, to show that treatment and response are independent. d. b and c.

Answer 1

Variables used for the following meaning :-

x1 or x2= number of people who improved under treatment 1 or2
p1 or p2= population proportion of patients who improve under treatment 1 or 2
phat1 or phat2= sample proportion of patients who improve under treatment 1 or2
n1 or n2= sample size of treatment 1 or 2

qhat = average of phat1 and phat2

alpha = 0.05 is the significance level

The hypothesis can be stated as follows:-

H0: p1+=+p2

H1: p1+%3C%3E+p2

for a two tailed test. The rule is that

if the test statistic is between the two critical values, then we do not reject the null. If the test statistic is not between the two critical values, then we reject H0.

Treatment 1: 14 patients show improvement
so x1 = 14
This is out of 22+14+14 = 50 total
14/50 = 0.28 = 28% of patients show improvement for treatment 1
phat1 = 0.28

Treatment 2: 21 patients show improvement
so x2 = 21
This is out of 14+15+21 = 50 total
21/50 = 0.42 = 42% of patients show improvement for treatment 2
phat2 = 0.42

The sample sizes of each treatment group is 50, so
n1 = 50
n2 = 50

Let's find the "average" sample proportion value and call this qhat

qhat = (x1+x2)/(n1+n2)
qhat = (14+21)/(50+50)
qhat = 0.35

Using ghat the standard error can be find out as follows:-

SE = sqrt(qhat*(1-qhat)*(1/n1 + 1/n2))
SE = sqrt(0.35*(1-0.35)*(1/50 + 1/50))
SE = 0.0953939201417

Now onto the test statistic

z = (phat1-phat2)/(SE)
z = (0.28 - 0.42)/(0.0953939201417)
z = -1.4675987714106
z = -1.47

The level of significance is 5%. Alpha = 0.05
confidence level = 0.95

Using a table we get

value 1.960

Since this is a two-tailed test, this means the critical values are -1.960 and 1.960

Since -1.47 is definitely between -1.960 and 1.960, this means we do not reject the null. We must conclude that p1 = p2. There isn't significant evidence to prove it wrong.

So the final answer is
B. There is Not significant evidence, alpha 0.05 to show that there is a difference in the proportion of patients who show improvements between treatments 1 and 2.

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