Chapter 10, Problem T10.11SPT

(a)

To determine

To construct and interpret a $95\%$ confidence interval for the difference in the true mean length of hospital stay for normothermic and hypothermic patients.

Answer to Problem T10.11SPT

We are $95\%$ confident that the mean population difference is between $-4.175\text{ and -1}\text{.025}$ .

Explanation of Solution

It is given in the question that:

  $\begin{array}{l}{\overline{x}}_1=12.1\\ {\overline{x}}_2=14.7\\ s_1=4.4\\ s_2=6.5\\ n_1=104\\ n_2=96\\ c=95\%\end{array}$

Now calculating the degree of freedom:

  $\begin{array}{l}df=\min (n_1-1,n_2-1)\\ =\min (104-1,96-1)\\ =95>80\end{array}$

Now, finding $t_c$ with $df=80$ and $c=95\%$ using table B:

  $t_c=1.990$

The endpoints of the confidence interval for ${\mu }_1-{\mu }_2$ are:

  $\begin{array}{l}({\overline{x}}_1-{\overline{x}}_2)-t_{\alpha /2}\sqrt{\frac{s_1{}^2}{n_1}+\frac{s_2{}^2}{n_2}}\\ =(12.1-14.7)-1.990\times \sqrt{\frac{{4.4}^2}{104}+\frac{{6.5}^2}{96}}\\ =-4.175\\ ({\overline{x}}_1-{\overline{x}}_2)+t_{\alpha /2}\sqrt{\frac{s_1{}^2}{n_1}+\frac{s_2{}^2}{n_2}}\\ =(12.1-14.7)+1.990\times \sqrt{\frac{{4.4}^2}{104}+\frac{{6.5}^2}{96}}\\ =-1.025\end{array}$

Thus we are $95\%$ confident that the mean population difference is between $-4.175\text{ and -1}\text{.025}$ .

(b)

To determine

To find out does your interval in part (a) suggest that keeping patients warm during surgery affects the average length of patients’ hospital stay or not.

Answer to Problem T10.11SPT

Yes.

Explanation of Solution

The interval in part (a) suggest that keeping patients warm during surgery affects the average length of patients’ hospital stay, it is true because the confidence interval $(-4.175,-1.025)$ in part (a) does not contain zero and thus suggest that keeping patients warm during surgery affects the average length of the patient’s hospital stay.

(c)

To determine

To interpret the meaning of “ $95\%$ confidence” in the context of this study.

Explanation of Solution

The meaning of “ $95\%$ confidence” in the context of this study is that if we were to repeat this experiment many times, about $95\%$ of the intervals would capture the true difference in mean hospital stay for patients like these who get heating blankets during surgery and mean hospital stay for patients who have core temperature reduced during surgery.