A clinical psychologist wants to choose between two therapies for treating sever

Question

A clinical psychologist wants to choose between two therapies for treating severe cases of mental depression. She selects six patients who are similar in their depressive symptoms and in their overall quality of health. She randomly selects three of the patients to receive Therapy 1, and the other three receive Therapy 2. She selects small samples for ethical reasons— if her experiment indicates that one therapy is superior, she will use that therapy on all her other depression patients. After one month of treatment, the improvement in each patient is measured by the change in a score for measuring severity of mental depression. The higher the score, the better. The improvement scores are

Therapy 1: 30, 45, 45

Therapy 2: 10, 20, 30

Analyze these data using R software

A)Show that x1 = 40,x2 = 20,s = 9.35 , se = 7.64, df = 4, and a 95% confidence interval comparing the means is (-1.2, 41.2).

B)For the null hypothesis, H0: 1 = 2, show that t = 2.62 and the two-sided P@value = 0.059. Interpret.

C)What decision would you make in the test, using a (i) 0.05 and (ii) 0.10 significance level? Explain what this means in the context of the study.

D)Suppose the researcher had predicted ahead of time that Therapy 1 would be better. To which Ha does this correspond? Report the P-value for it, and make a decision with significance level 0.05.

Explanation / Answer

1)

x1 <- c(30,45,45)
> x2 <- c(10,20,30)
> mean (x1)
[1] 40
> mean (x2)
[1] 20
> sd(x1)
[1] 8.660254
> sd(x2)
[1] 10
> s = sqrt((var(x1) + var(x2))/2)
> s
[1] 9.354143

df = n1 + n2 - 2 = 4

qt(0.975,4)
[1] 2.776445

t0.025 = 2.776445

hence confidence interval is

(40-20) - 2.776445*9.354143 *sqrt(1/3+1/3) , ((40-20) +2.776445*9.354143 ,sqrt(1/3+1/3)

=( -1.205448,41.20545)

B) TS = (X1bar - X2bar) /s(e *sqrt(1/n1+1/n2) ) = 20/(9.35* sqrt(1/3+1/3)) =2.61977

p-value = 2 P(t> 2.61977) = 0.059

2*pt (-2.61977 ,df =4)
[1] 0.05881357

C) when p-value is less than level of significance ,

we reject the null hypothesis

i)p-value > 0.05 - We fail to reject the null hypothesis

ii) p-valua <0.10   We reject the null hypothesis

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