Question

Employees of a certain state university system can choose from among four different health plans. Each plan differs somewhat from the others in terms of hospitalization coverage. Four samples of recently hospitalized individuals were selected, each sample consisting of people covered by a different health plan. The length of the hospital stay (number of days) was determined for each individual selected. a. What hypotheses would you test to decide whether average length of stay was related to health plan? (Note: Carefully define the population characteristics of interest.) b. If each sample consisted of eight individuals and the value of the ANOVA \(F\) statistic was \(F=4.37\), what conclusion would be appropriate for a test with \(\alpha=.01\) ? c. Answer the question posed in Part (b) if the \(F\) value given there resulted from sample sizes \(n_{1}=9, n_{2}=8\), \(n_{3}=7\), and \(n_{4}=8\).

Short answer

H{0}: There is no difference in the average length of stay between different health plans. H{A}: There is a difference in the average length of stay among the health plans. Without more specific information such as the number of total observations and sum of squares, it is impossible to conclusively interpret the F-statistic or to adjust the process for differing sample sizes.

Step-by-step solution

Formulate the hypotheses

For an analysis of variance (ANOVA), there are always two hypotheses: \n\n1. Null Hypothesis (\(H{0}\)): The means of the different health plans are equal. This implies that there is no difference in the average length of stay among different health plans. \n\n2. Alternative Hypothesis (\(H{A}\)): At least one of the means of the health plans is different. This suggests that there is a difference in the average length of stay among the different health plans.

Interpreting the F-statistic for the given parameters

Given that the value of the F-statistic is 4.37 and alpha level (\(\alpha\)) is 0.01, there is a need to compute the critical value of F for the given alpha level and degrees of freedom. If F calculated (4.37) is greater than the F critical, reject the null hypothesis. However, the problem did not provide the degrees of freedom needed to calculate the F critical value - it is usually calculated as k-1 (where k is number of groups) and N-k (where N is total number of observations). Without the total number of observations, cannot compute the F critical value

Analysis with different sample sizes

For the third question, we are given different sample sizes for each group (\(n_{1}=9, n_{2}=8, n_{3}=7, and n_{4}=8\)) and the same F-value (4.37). But without the total sum of squares within the groups and between groups, it is impossible to estimate the variability of the data. It remains an incomplete information problem.

Key concepts

Hypothesis Testing in ANOVA

Hypothesis testing is a fundamental concept in statistics that allows researchers to make inferences about population parameters based on sample data. In the context of health plan analysis, as seen in the exercise, hypothesis testing via ANOVA aims to determine whether the average length of hospital stay is influenced by the type of health plan participants have.

When employing ANOVA for hypothesis testing, the null hypothesis (\( H_0 \)), asserts that the means across all groups are equal. Specifically, in our health plan example, \( H_0 \) posits that the different health plans do not affect the average length of hospital stay. Conversely, the alternative hypothesis (\( H_A \)) suggests that at least one health plan's mean length of stay is significantly different from the others.

To test these hypotheses, ANOVA computes an F-statistic, which measures the ratio of the variability between group means to the variability within the groups. If this F-statistic is significantly high, it indicates that the group means are not all identical in the population, prompting researchers to reject the null hypothesis. A critical value is determined based on the desired alpha level (\( \)alpha)), which represents the threshold for significance. Investigators conclude there is a significant effect if the calculated F exceeds this critical value.

F-Statistic Interpretation

The F-statistic is a key measure in ANOVA that helps in determining whether the group means are statistically different from each other. As shown in the exercise, the computed F-statistic was 4.37. To interpret this, one would compare it to a critical value which is chosen based on the significance level (\( \)alpha)) and the degrees of freedom for the numerator (between-groups variability) and the denominator (within-groups variability).

For the given significance level of \( alpha=.01 \) , we would typically consult an F-distribution table or use a software calculator to determine the critical F-value. If our calculated F of 4.37 is greater than the critical F-value, it would suggest that at least one group mean is significantly different from the others, leading to the rejection of the null hypothesis.

However, it's important to note that the F-statistic must be interpreted within the context of its degrees of freedom which are derived from the number of groups and the total sample size. Since the exact degrees of freedom were not provided in the exercise, we proceed under the assumption that the computed F-statistic of 4.37 is indeed greater than the critical value, indicating significant differences between the health plans regarding the average length of the hospital stay.

Sample Size Impact on ANOVA

Sample size plays a crucial role in ANOVA and impacts the robustness and sensitivity of the test. Larger sample sizes generally provide more accurate estimates of the population parameters and increase the power of the test, which is the likelihood of correctly rejecting a false null hypothesis.

In the exercise, we observe differing sample sizes for each health plan group. These dissimilarities in sample sizes can affect the ANOVA results. With equal or balanced sample sizes, the test has maximum power, but unbalanced sample sizes can result in a loss of power and complicate the interpretation of results.

Assuming the F-statistic remains the same at 4.37 with the altered group sizes as mentioned in part (c), we must still be cautious. The varying sample sizes have also changed the degrees of freedom, which subsequently alter the critical values needed for comparing against the calculated F-statistic. This requires a recalculation of the F critical value to determine if the observed F-statistic is still indicative of a significant difference between group means.

In conclusion, the impact of varying sample sizes in ANOVA necessitates a thorough analysis, and researchers must account for these differences when drawing conclusions about the population from sample data.