Final answer:
The hypothesis test suggests no significant difference in the mean length of hospital stays for pneumonia patients compared to the national average of 5.4 days. The p-value exceeds the 0.05 significance level, leading to the failure to reject the null hypothesis.
Explanation:
In conducting the hypothesis test to determine if the mean length of hospital stays for pneumonia patients differs from the national average of 5.4 days, a sample of 40 patients revealed a mean length of 5.2 days with a standard deviation of 1.1 days. Employing a significance level (alpha) of 0.05, we established the null hypothesis (H0) that the mean length is equal to 5.4 days and the alternative hypothesis (H1) that it is different.
The calculated p-value, which represents the probability of obtaining the observed results under the assumption that the null hypothesis is true, was compared to the significance level. The results indicated a p-value exceeding 0.05, signifying that the observed difference in mean lengths is not statistically significant.
Consequently, we fail to reject the null hypothesis, suggesting insufficient evidence to conclude a difference in mean hospital stays for pneumonia patients compared to the national average. This outcome implies that, within the bounds of the sample data, the mean length of hospital stays for pneumonia patients appears consistent with the national average of 5.4 days.