 # The following data, recorded in days, represents the recovery time, for patients he298c 2021-09-21 Answered

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:

Find the $99\mathrm{%}$ confidence interval for $\mu 1-\mu 2$, the difference in mean drug recovery times, and INTERPRET it to get a helpful conclusion about the drugs.
Assume normal populations, with equal variances.

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