Question

A hospital administrator is interested to compare the average hospital stay at three hospitals in a certain city. The administrator collects random data for the length of hospital stay (in days) for the three hospitals and the dataset is provided below. The administrator is interested to know if the average hospital stays are statistically the same for the three hospitals. Use a significance level of 5%. The administrator has confirmed that the samples were randomly selected and independent, and the populations have normal distribution and the population variances are equal.

a. Calculate the Test Statistic for this example (round your answer to 2 decimal places)

b. Calculate the P-value for this example (round your answer to 2 decimal places)


Hospital A Hospital B Hospital C

4.4 5.4 7.6

1.1 4.2 5.4

4.9 3.7 6.9

2.6 5.6 2.6

4.9 5.9 5.4

3.9 4.4 6.7

3.9 3.8 7.9

Answer 1

The ANOVA test is used to determine if there is a statistical difference in the means of the hospital stay lengths among three hospitals, using a significance level of 0.05. The steps include calculating means, variances, sum of squares, and finally the F-test statistic, followed by determining the p-value to make a decision on the null hypothesis.

Step-by-step explanation:

To determine whether the average hospital stays are statistically the same for the three hospitals, we would use an ANOVA (Analysis of Variance) test, as we are comparing the means of three independent groups and the assumptions of normality, independence, and equal variances are met. The significance level (alpha) is set at 0.05. Without the dataset itself, we cannot calculate the exact test statistic or p-value here. However, I can provide a general outline of the steps taken to perform this test:

1. Calculate the mean and variance for each of the three hospitals.
2. Calculate the overall mean (grand mean) of all the data combined.
3. Determine the Between Group Sum of Squares (SSB) which measures the variability of the group means around the grand mean.
4. Calculate the Within Group Sum of Squares (SSW) which measures the variability within each group.
5. Determine the total Sum of Squares (SST) which is the sum of SSB and SSW.
6. Calculate the Between Group Mean Square (MSB) by dividing SSB by the between group degrees of freedom (number of groups minus one).
7. Calculate the Within Group Mean Square (MSW) by dividing SSW by the within group degrees of freedom (total number of observations minus the number of groups).
8. The F-test statistic is then found by dividing MSB by MSW.
9. Finally, we compare the computed F-test statistic to the F-distribution at the degrees of freedom associated with MSB and MSW to determine the p-value.
10. If the p-value is less than the significance level of 0.05, we reject the null hypothesis, indicating that there is a statistically significant difference between the means of the three hospitals.

For more precise calculations, appropriate statistical software or a calculator function can be used to compute the test statistic and p-value.