The National Coalition on Healthcare suggests that the mean annual premium that a health insurer charges an employer for a health plan covering a family of four averaged $12700 in 2008. A sample of 30 families of four yield a mean annual premium paid by their employer to be $13200 with a sample standard deviation of $300. We are interested in whether the mean annual premium that a health insurer charges an employer for a health plan covering a family of four is different from $12700 using a significance level of 0.10.

Question

The National Coalition on Healthcare suggests that the mean annual premium that a health insurer charges an employer for a health plan covering a family of four averaged $12700 in 2008. A sample of 30 families of four yield a mean annual premium paid by their employer to be $13200 with a sample standard deviation of $300. We are interested in whether the mean annual premium that a health insurer charges an employer for a health plan covering a family of four is different from $12700 using a significance level of 0.10.

Answer 1

To determine whether the mean annual premium charged by a health insurer for a health plan covering a family of four is significantly different from $12,700, we can perform a hypothesis test.

Here's how you can conduct the hypothesis test:

Step 1: Formulate the null and alternative hypotheses:
- Null Hypothesis (H0): The mean annual premium charged by a health insurer is equal to $12,700 (µ = $12,700).
- Alternative Hypothesis (Ha): The mean annual premium charged by a health insurer is different from $12,700 (µ ≠ $12,700).

Step 2: Define the significance level (α):
Given that the significance level is 0.10, this means we have a 10% chance of rejecting the null hypothesis when it is actually true.

Step 3: Calculate the test statistic:
The test statistic for comparing means when the population standard deviation is unknown is the t-test statistic. We can calculate it using the following formula:

t = (x̄ - µ) / (s / √n)

Where:
- x̄ is the sample mean ($13,200)
- µ is the population mean ($12,700)
- s is the sample standard deviation ($300)
- n is the sample size (30)

Calculating the t-test statistic:

t = (13,200 - 12,700) / (300 / √30) ≈ 2.04 (rounded to two decimal places)

Step 4: Determine the critical value(s):
Since we are performing a two-tailed test, we need to find the critical value(s) for a significance level of 0.10 by dividing the significance level by 2 (0.10 / 2 = 0.05). Looking up the critical value in the t-distribution table with 29 degrees of freedom and a significance level of 0.05 (two-tailed), we find the critical value to be approximately ±1.699.

Step 5: Compare the test statistic with the critical value(s):
If the absolute value of the test statistic is greater than the critical value, we can reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, |2.04| > 1.699, so we can reject the null hypothesis.

Step 6: Draw a conclusion:
Based on the results, we can conclude that there is sufficient evidence to suggest that the mean annual premium charged by a health insurer for a health plan covering a family of four is different from $12,700 at a significance level of 0.10.

To summarize, by calculating the t-test statistic and comparing it with the critical value, we can determine if the mean annual premium is significantly different from the given value.