Question

The Providence Heart Institute in Columbia, South Carolina, performs many open-heart surgery procedures. Recently, research physicians at Providence have developed a new heart bypass surgery procedure that they believe will reduce the average recovery time. The hospital board will not recommend the new procedure unless there is substantial evidence to suggest that it is better than the existing procedure. Records indicate that the current mean recovery rate for the standard procedure is 42 days, with a standard deviation of 5 days. To test whether the new procedure actually results in a lower mean recovery time, the procedure was performed on a random sample of 36 patients and found the mean recovery rate of 40 days. (Use p-value method.)

Answer 1

Let be the true mean recovery time for the new heart bypass surgery procedure. We want to test whether the new procedure actually results in a lower mean recovery time, that is if the mean recovery time for new procedure is less than the recovery time for standard procedure (which is 42 days). That means we want to test if

The following hypotheses need to be tested

We have the folowing information from the sample

n=36 is the sample size of the patients on who the new procedure was performed

is the sample mean recovery time for the new procedure

is the population standard deviation of recovery time

is the standard error of mean

is the hypothesized value of mean recovery time

The sample size n is greater than 30 and also we know the population standard deviation. That means normal distribution will be used to test the hypotheses

The test statistics is

This is a one tail (left tail, as the alternative hypothesis is <42) test

The p-value is the area under the left tail. That is the p-value is P(Z<-2.4)=P(Z>2.4) = 1-P(Z<2.4)

Uisng the standard normal tables we get for z=2.4 P(Z<2.4) = 0.5+0.4918=0.9918

Hence p-value = 1-0.9918 = 0.0082

We will reject the null hypothesis if the p-value is less than the significance level alpha=0.05.

This p-value is less than the significance level alpha=0.05. Hence we will reject the null hypothesis.

We conclude that there is sufficient evidence to support the claim that  the new procedure actually results in a lower mean recovery time.