Late payment of medical claims can add to the cost of health care. An article (M. Freudenheim, "The Check Is Not in the Mail," The New York Times, May 25, 2006, pp. C1, C6) reported that for one insurance company, 85.1% of the claims were paid in full when first submitted. Suppose that the insurance company developed a new payment system in an effort to increase this percentage. A sample of 200 claims processed under this system revealed that 180 of the claims were paid in full when first submitted. a. At the 0.05 level of significance, is there evidence that the population proportion of claims processed under this new system is higher than the article reported for the previous system? b. Compute the p-value and interpret its meaning.

Question

Late payment of medical claims can add to the cost of health care. An article (M. Freudenheim, "The Check Is Not in the Mail," The New York Times, May 25, 2006, pp. C1, C6) reported that for one insurance company, 85.1% of the claims were paid in full when first submitted. Suppose that the insurance company developed a new payment system in an effort to increase this percentage. A sample of 200 claims processed under this system revealed that 180 of the claims were paid in full when first submitted. a. At the 0.05 level of significance, is there evidence that the population proportion of claims processed under this new system is higher than the article reported for the previous system? b. Compute the p-value and interpret its meaning.

Answer 1

To answer this question, we will conduct a hypothesis test for the proportion and compare the sample proportion to the reported proportion from the article.

a. Hypotheses:
- Null hypothesis (H0): The population proportion for the new payment system is the same as the reported proportion for the previous system (p = 0.851).
- Alternative hypothesis (Ha): The population proportion for the new payment system is higher than the reported proportion for the previous system (p > 0.851).

b. Test statistic:
For this hypothesis test, we will use the Z-test for proportions. The test statistic is calculated using the formula:

Z = (p̂ - p) / sqrt((p * (1 - p)) / n)

where p̂ is the sample proportion, p is the population proportion from the article, and n is the sample size.

In this case, p̂ = 180/200 = 0.9, p = 0.851, and n = 200. Plugging these values into the formula, we can calculate the test statistic.

Z = (0.9 - 0.851) / sqrt((0.851 * (1 - 0.851)) / 200) = 1.792

c. Significance level and critical value:
The significance level is given as 0.05. Since this is a one-tailed test (we are testing if the proportion is higher than the reported proportion), we need to find the critical value for a one-tailed test with an alpha level of 0.05.

Looking up the critical value in the Z-table, the critical value for a one-tailed test at a 0.05 significance level is approximately 1.645.

d. Decision rule:
If the test statistic falls in the critical region (greater than the critical value), we will reject the null hypothesis. Otherwise, we will fail to reject the null hypothesis.

e. Calculation of the p-value:
The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. To calculate the p-value, we look up the Z-score in the standard normal distribution table.

The p-value for a test statistic of 1.792 is 0.0364.

f. Interpretation:
Since the test statistic (1.792) is greater than the critical value (1.645) and the p-value (0.0364) is less than the significance level (0.05), we reject the null hypothesis.

Therefore, there is evidence to suggest that the population proportion of claims processed under the new payment system is higher than the reported proportion for the previous system.

Please note that this interpretation assumes all assumptions for the Z-test for proportions are met, such as the sample being representative of the population, and the samples being independent.