Question

Acute Postoperative Days. Refer to Example $10.6$(page 420). The researchers also obtained the following data on the number of acute postoperative days in the hospital using the dynamic and static systems.

At the $5\%$significance level, do the data provide sufficient evidence to conclude that the mean number of acute postoperative days in the hospital is smaller with the dynamic system than with the static system? (Note: $x\_\left\{1\right\}=7.36,$$s\_\left\{1\right\}=1.22,$$x\_\left\{2\right\}=10.50$and $s\_\left\{2\right\}=4.59.$)

Short answer

The statistics give adequate evidence to conclude that the two population means are equivalent at the significance level of $5\%$. The data does not give sufficient evidence to establish that the mean the number of post - operative days in the hospital is lower with the dynamic system than with the static system at the $5\%$ significance level.

Step-by-step solution

Given Information 

Using the critical value strategy or the $p$value approach, determine the required hypothesis.

Explanation 

The hypothesis test to be carried out is as follows:

$H_{\mathrm{a}}:{\mu }_1<{\mu }_2$

The test is run at a significance level of $\alpha =0.05$which is equal to $5\%$.

The test statistic's value is calculated as follows:

The value of test statistic is calculated as,

$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\left(\frac{s_1^2}{n_1}\right)+\left(\frac{s_2^2}{n_1}\right)}}$

$=\frac{7.36-10.50}{\sqrt{\left(\frac{1.{22}^2}{14}\right)+\left(\frac{4.{59}^2}{6}\right)}}$

$=-1.65$

Step 3: 

The$t$statistic has $df=\Delta$

The probability of seeing a value$t=-1.65$is given by the localid="1651290852925" $P$value.

The value of localid="1651290856971" $P$obtained using technological means is localid="1651290861835" $0.0798$.

The significance level of localid="1651290866001" $1\%$is exceeded by the localid="1651290870640" $P$value.

As a result, at the localid="1651290901161" $5\%$level, the results are not statistically significant.