Question

The following data, recorded in days, represents the recovery time, for patients who are randomly treated with one of two medications to cure servere bladder infections:
\(\begin{array}{|c|c|}\hline Medication\ 1 & Medication\ 2 \\ \hline n_{1}=13 & n_{2}=16 \\ \hline \bar{x}_{1}=20 & \bar{x}_{2}=15 \\ \hline s_{1}^{2}=2.0 & s_{2}^{2}=1.8 \\ \hline \end{array}\)
Find the \(\displaystyle{99}\%\) confidence interval for \(\displaystyle\mu{1}-\mu{2}\), the difference in mean drug recovery times, and INTERPRET it to get a helpful conclusion about the drugs.
Assume normal populations, with equal variances.

Answer 1

Step 1
Given data:
\(\displaystyle{n}_{{{1}}}={13}\),
\(\displaystyle\overline{{{x}}}_{{{1}}}={20}\)
\(\displaystyle{\sigma_{{{1}}}^{{{2}}}}={1}\),
\(\displaystyle\sigma_{{{1}}}={1}\)
\(\displaystyle{n}_{{{2}}}={16}\),
\(\displaystyle\overline{{{x}}}_{{{2}}}={15}\),
\(\displaystyle{\sigma_{{{2}}}^{{{2}}}}={1.8}\)
\(\displaystyle\sigma_{{{2}}}={1.34}\)
Confidence level \(\displaystyle={99}\%\)
The formula for confidence interval is:
\(\displaystyle{C}.{I}.=\overline{{{X}}}\pm{Z}\times{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)
For the sample 1:
Put the values for sample 1:
\(\displaystyle{C}.{I}.=\overline{{{X}}}\pm{Z}\times{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)
\(\displaystyle{C}.{I}.={20}\pm{2.5758}\times{\frac{{{1}}}{{\sqrt{{{13}}}}}}\)
\(\displaystyle{C}.{I}.={20}\pm{0.714}\)
\(\displaystyle{C}.{I}.={\left[{19.28}-{2.71}\right]}\)
Step 2
For sample 2 the CI is:
\(\displaystyle{C}.{I}.=\overline{{{X}}}\pm{Z}\times{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)
\(\displaystyle{C}.{I}.={15}\pm{2.5758}\times{\frac{{{1.34}}}{{\sqrt{{{16}}}}}}\)
\(\displaystyle{C}.{I}.={15}\pm{0.863}\)
\(\displaystyle{C}.{I}.={\left[{14.14}-{15.86}\right]}\)
For \(\displaystyle\mu_{{{1}}}-\mu_{{{2}}}\):
\(\displaystyle\mu_{{{1}}}-\mu_{{{2}}}=\overline{{{X}}}_{{{1}}}-\overline{{{X}}}_{{{2}}}\pm{Z}{\left(\sqrt{{{\frac{{{\sigma_{{{1}}}^{{{2}}}}}}{{{n}_{{{1}}}}}}-{\frac{{{\sigma_{{{2}}}^{{{2}}}}}}{{{n}_{{{2}}}}}}}}\right)}\)
\(\displaystyle\mu_{{{1}}}-\mu_{{{2}}}={20}-{15}\pm{2.77}{\left({0.45}\right)}\)
\(\displaystyle\mu_{{{1}}}-\mu_{{{2}}}={5}\pm{1.24}\)
\(\displaystyle\mu_{{{1}}}-\mu_{{{2}}}={\left[{3.76}-{6.24}\right]}\)