A medical billing specialist believes that most patients with high-deductible health plans have overdue medical bills. The medical billing specialist would like to test the claim that the proportion of patients with high-deductible health plans who have overdue medical bills is greater than 51%. They sample 35 patients with high-deductible health plans and determine the sample percent to be 60%. a) State the null hypothesis (H₀). b) State the alternative hypothesis (H₁). c) Choose a significance level (alpha, α). d) Calculate the test statistic. e) Determine the critical value. f) Decide whether to reject or fail to reject the null hypothesis. g) State your conclusion.

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Answer 1

Answer:

The medical billing specialist wants to test whether the proportion of patients with high-deductible health plans who have overdue medical bills is greater than 51%.

a. Null Hypothesis (H₀): H₀: p ≤ 0.51

b. Alternative Hypothesis (H₁): H₁: p > 0.51

c. Significance Level (alpha, α): α = 0.05.

d. Calculate the Test Statistic: $z = \frac{0.60 - 0.51}{\sqrt{(0.51(1-0.51))/(35)}}$

e. Determine the Critical Value: approximately 1.645

f. If (z > 1.645), you will reject the null hypothesis; otherwise, you will fail to reject it.

g. Conclusion: If the test statistic is greater than 1.645, you can conclude that there is sufficient evidence to support the claim that more than 51% of patients with high-deductible health plans have overdue medical bills. If the test statistic is less than 1.645, you would not have enough evidence to support this claim.

The medical billing specialist wants to test whether the proportion of patients with high-deductible health plans who have overdue medical bills is greater than 51%. Let's go through the steps of hypothesis testing:

Null Hypothesis (H₀): The null hypothesis states that there is no significant difference, and the proportion of patients with high-deductible health plans who have overdue medical bills is equal to or less than 51%.

H₀: p ≤ 0.51

Alternative Hypothesis (H₁): The alternative hypothesis is the claim the specialist wants to test, which is that the proportion of patients with overdue medical bills is greater than 51%.

H₁: p > 0.51

Significance Level (alpha, α): The significance level represents the level of risk you are willing to take for making a Type I error (rejecting the null hypothesis when it's true). Common values are 0.05 or 0.01. Let's choose α = 0.05.

Calculate the Test Statistic: You can use the sample proportion and standard error to calculate the test statistic, which follows a z-distribution:

$z = \frac{\hat{p} - p}{\sqrt{(p(1-p))/(n)}}$

Where:

- $\hat{p}$ is the sample proportion (0.60).

- (p) is the proportion under the null hypothesis (0.51).

- (n) is the sample size (35).

Calculating (z):

$z = \frac{0.60 - 0.51}{\sqrt{(0.51(1-0.51))/(35)}}$

Determine the Critical Value: At α = 0.05, using a one-tailed test (since we're testing whether it's greater than 51%), the critical value is approximately 1.645 (you can find this from a standard normal distribution table).

Decision: Compare the calculated test statistic (step d) with the critical value (step e). If (z > 1.645), you will reject the null hypothesis; otherwise, you will fail to reject it.

Conclusion: If the test statistic is greater than 1.645, you can conclude that there is sufficient evidence to support the claim that more than 51% of patients with high-deductible health plans have overdue medical bills. If the test statistic is less than 1.645, you would not have enough evidence to support this claim.

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