The U.S. Department of Labor collects data on unemployment insurance payments. Suppose that during 2009 a random sample of 80 unemployed people in Alabama received an average weekly benefit of $201.85, whereas a random sample of 65 unemployed people in Mississippi received an average weekly benefit of $185.31. Assume that the population standard deviations of all weekly unemployment benefits in Alabama and Mississippi are $32.48 and $26.15, respectively. Using the 4% significance level, can you conclude that the means of all weekly unemployment benefits in Alabama and Mississippi paid during 2009 are different?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of Z score to compare with .04
To determine whether the means of all weekly unemployment benefits in Alabama and Mississippi are different, we can perform a two-sample t-test. This test allows us to compare the means of two samples from different populations. Here's how you can do it:
Step 1: State the hypotheses:
- Null hypothesis (H0): The means of all weekly unemployment benefits in Alabama and Mississippi are the same.
- Alternative hypothesis (H1): The means of all weekly unemployment benefits in Alabama and Mississippi are different.
Step 2: Determine the significance level:
In this case, the significance level is given as 4%. This means that we will reject the null hypothesis if the p-value calculated from the test is less than 0.04.
Step 3: Conduct the t-test:
The formula for the two-sample t-test is:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
Where:
x1 and x2 are the sample means (from Alabama and Mississippi, respectively),
s1 and s2 are the sample standard deviations (from Alabama and Mississippi, respectively),
n1 and n2 are the sample sizes (from Alabama and Mississippi, respectively).
Step 4: Calculate the test statistic:
Using the given information:
x1 = $201.85, x2 = $185.31,
s1 = $32.48, s2 = $26.15,
n1 = 80, n2 = 65,
t = (201.85 - 185.31) / sqrt((32.48^2 / 80) + (26.15^2 / 65))
Step 5: Determine the p-value and make a decision:
Using the calculated test statistic, we can determine the p-value associated with it. The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level (0.04), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
You can use statistical software or a t-table to find the p-value associated with the calculated t-value. By comparing the p-value to the significance level, you can conclude whether the means of all weekly unemployment benefits in Alabama and Mississippi are different or not.
Note: If you have statistical software such as R or Python, you can directly input the sample data to conduct the t-test and obtain the p-value.