Question

A new post-surgical treatment is being compared with a standard treatment. Seven subjects receive the new treatment, while seven others (the controls) receive the standard treatment. The recovery times, in days, are given below.

Treatment: 12 13 15 19 20 21 24
Control: 18 23 24 30 32 35 39

Required:
Find a 98% confidence interval for the difference in the mean recovery times between treatment and control.

Answer 1

`(17.714-28.714) -2.681 \sqrt{(4.461^2)/(7) +(7.387^2)/(7)}= -19.745`

`(17.714-28.714) +2.681 \sqrt{(4.461^2)/(7) +(7.387^2)/(7)}= -2.255`

Explanation:

For this case we have the following info given:

Treatment: 12 13 15 19 20 21 24

Control: 18 23 24 30 32 35 39

We can find the sample mean and deviations with the the following formulas:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

`s =\sqrt{(\sum_(i=1)^n (X_i- \bar X)^2)/(n-1)}`

And repaplacing we got:

`\bar X_T = 17.714` the sample mean for treatment

`\bar X_C = 28.714` the sample mean for treatment

`s_T= 4.461` the sample deviation for treatment

`s_C= 7.387` the sample deviation for control

`n_T= n_C= 7` the sample size for each sample

The degrees of freedom are given by:

`df= 7+7-2= 12`

The confidence interval for the difference of means is given by:

`(\bar X_T -\bar X_C) \pm t_(\alpha/2) \sqrt{(s^2_T)/(n_T) +(s^2_C)/(n_C)}`

The confidence is 98% so then the significance is
`\alpha=0.02` and
`\alpha/2 =0.01`. Then the critical value would be:

`t_(\alpha/2)=2.681`

And replacing we got:

`(17.714-28.714) -2.681 \sqrt{(4.461^2)/(7) +(7.387^2)/(7)}= -19.745`

`(17.714-28.714) +2.681 \sqrt{(4.461^2)/(7) +(7.387^2)/(7)}= -2.255`