Question

2. A medical group is developing a new diabetes therapy that they hope will lower a patient's lc measurement. They have a control group of 137 patients with a mean Alc of 8.43 and a sample standard deviation of Sc-1.78. The researcher would like to compare these numbers o a treatment group. The Alc measurements for the 5 subjects in the treatment group were 4.9 5.2 6.8 8.7 9.1 We will assume that these measurements are independent and the variance in both groups is the same. We want to test if the mean of Alc measurements in the treatment group is significantly lower. (a) Calculate an estimate of the variance of the Alc measurements. (b) Calculate the appropriate test statistic for this test. (c) What is the distribution of the test statistic under the null hypothesis? How many degrees of freedom does it have? (d) Use a large sample approximation to find a P value for this test. (e) What do we conclude about the effectiveness of the medication?

Answer 1

$(a)~Estimate~of~the~variance=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}=3.1849\\ Here,~n_1=sample~size~of~control~group=137,~s_1=sample~sd~of\\ ~control~group=1.78,~n_2=sample~size~of~treatment~group=5,\\ s_2=sample~sd~of~treatment~group=1.935\\ (b)~Let~\mu_1=true~mean~of~A1e~measurement~for~control~group\\ \mu_2=true~mean~of~A1e~measurement~for~treatment~group.\\ Null~hypothesis,~H_0:\mu_1=\mu_2~vs.~Alternative~hypothesis,~H_1:\mu_1>\mu_2.\\\\ Test~statistic=t=\frac{\bar{x}_1-\bar{x}_2}{s\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$where,~\bar{x}_1=sample~mean~for~control~group=8.43~and~\bar{x}_2=sample~mean~of\\ treatment~group=6.94.\\ Value~of~t=1.0275\\ (c)~Under~null~hypothesis,~the~distribution~of~t~is~t-distribution~with~\\ 137+5-2=140~df.\\ (d)~pp-value=P(t>1.0275|t\sim t_{140})=0.1530.\\ (e)~Since~p-value>0.05,~so~we~fail~to~reject~H_0~at~5\%~level~of~significance~and~conclude~that\\ there~is~insufficient~evidence~to~conclude~that~mean~of~A1c~measurements\\ in~treatment~group~is~significantly~lower.$