To clarify, the test statistic value for the data given is the F-statistic value calculated during the one-way ANOVA. Let's ensure the calculations are accurate by repeating the necessary steps.

Recalculation for Accuracy

Recovery Times for Each Treatment Group

• Sent Home With Medicine: 16, 16, 26, 24, 24, 30, 16, 20
• A Few Hours in the Hospital: 20, 13, 16, 9, 10, 17, 19, 12
• Overnight Hospital Stay: 18, 11, 10, 9, 10, 18, 6, 17, 24

Step 1: Calculate the Means for Each Group

$\bar{X}_1 = \frac{16 + 16 + 26 + 24 + 24 + 30 + 16 + 20}{8} = \frac{172}{8} = 21.5$ $\bar{X}_2 = \frac{20 + 13 + 16 + 9 + 10 + 17 + 19 + 12}{8} = \frac{116}{8} = 14.5$ $\bar{X}_3 = \frac{18 + 11 + 10 + 9 + 10 + 18 + 6 + 17 + 24}{9} = \frac{123}{9} \approx 13.67$

Step 2: Calculate the Overall Mean

$\bar{X} = \frac{\sum X}{N} = \frac{172 + 116 + 123}{25} = \frac{411}{25} = 16.44$

Step 3: Calculate the Sum of Squares Between Groups (SSB)

$SSB = \sum n_i (\bar{X}_i - \bar{X})^2$

$SSB = 8(21.5 - 16.44)^2 + 8(14.5 - 16.44)^2 + 9(13.67 - 16.44)^2$

$SSB = 8(5.06)^2 + 8(-1.94)^2 + 9(-2.77)^2$

$SSB = 8(25.6036) + 8(3.7636) + 9(7.6729)$

$SSB = 204.8288 + 30.1088 + 69.0561 = 303.9937$

Step 4: Calculate the Sum of Squares Within Groups (SSW)

$SSW = \sum (X_{ij} - \bar{X}_i)^2$

Calculate each group's sum of squares within:

Sent Home With Medicine: $SSW_1 = (16 - 21.5)^2 + (16 - 21.5)^2 + (26 - 21.5)^2 + (24 - 21.5)^2 + (24 - 21.5)^2 + (30 - 21.5)^2 + (16 - 21.5)^2 + (20 - 21.5)^2 = 250.5$

A Few Hours in the Hospital: $SSW_2 = (20 - 14.5)^2 + (13 - 14.5)^2 + (16 - 14.5)^2 + (9 - 14.5)^2 + (10 - 14.5)^2 + (17 - 14.5)^2 + (19 - 14.5)^2 + (12 - 14.5)^2 = 170.5$

Overnight Hospital Stay: $SSW_3 = (18 - 13.67)^2 + (11 - 13.67)^2 + (10 - 13.67)^2 + (9 - 13.67)^2 + (10 - 13.67)^2 + (18 - 13.67)^2 + (6 - 13.67)^2 + (17 - 13.67)^2 + (24 - 13.67)^2 = 427.56$

$SSW = 250.5 + 170.5 + 427.56 = 848.56$

Step 5: Calculate the Mean Square Between Groups (MSB)

$MSB = \frac{SSB}{k-1} = \frac{303.9937}{3-1} = \frac{303.9937}{2} = 151.99685$

Step 6: Calculate the Mean Square Within Groups (MSW)

$MSW = \frac{SSW}{N-k} = \frac{848.56}{25-3} = \frac{848.56}{22} = 38.5718$

Step 7: Calculate the F-statistic

$F = \frac{MSB}{MSW} = \frac{151.99685}{38.5718} \approx 3.94$

Conclusion

The test statistic value (F-statistic) for the given data is approximately 3.94.

Do you want more details or have any questions?

Here are 5 questions you might want to ask next:

1. How is the critical value for the F-statistic determined?
2. What are the assumptions of the ANOVA test?
3. How can we conduct a post-hoc test to determine which means are different?
4. Can you explain the concept of degrees of freedom in this context?
5. How would we handle the analysis if the standard deviations were not the same?

Tip: Ensure the normality and equal variance assumptions hold before interpreting the results of an ANOVA test.