2. In 2009 a random sample of 70 unemployed people in Alabama showed an average weekly benefit of $199.65. In Mississippi, for a random sample of 65 the number was $187.93. Assume population standard deviations of $32.48 and $26.15 respectively.
a. Using the 5% level of significance, test whether
the two means are different.
b. Assume the p-value for this test is .0439.
c. If the level of significant used was 10% (or 1%),
would the Ho be rejected?
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 related to the Z score.
a. Well, it seems like Alabama's unemployed folks are making it rain with their average weekly benefit of $199.65, while Mississippi is keeping it a little more modest with $187.93. But are these means statistically different? Let's find out!
b. Ah, the p-value, that magical little number that determines whether we reject the null hypothesis. In this case, the p-value is 0.0439. So if we're using the 5% level of significance, which is our standard threshold for statistical significance, this p-value is actually less than 0.05. So we can reject the null hypothesis and confidently say that the means are different. Goodbye, boring equal means, hello exciting unequal means!
c. Now, if we were to use a higher level of significance, say 10% or 1%, would we still reject the null hypothesis? Well, if we were using the 10% level of significance, which is a more lenient threshold, we would still reject the null hypothesis because our p-value of 0.0439 is less than 0.10. However, if we were using the 1% level of significance, which is a lot more strict, we would NOT reject the null hypothesis because our p-value is greater than 0.01. So it all depends on how stringent we want to be with our statistical decisions.
a. To test whether the two means are different, we can perform a two-sample t-test. The null hypothesis (Ho) is that there is no difference in the means of weekly benefits between Alabama and Mississippi unemployed people. The alternative hypothesis (Ha) is that there is a difference in the means.
Step 1: Set up the hypotheses:
Ho: μ1 = μ2 (the means are equal)
Ha: μ1 ≠ μ2 (the means are different)
Step 2: Determine the significance level (α). In this case, the significance level is 5% or 0.05.
Step 3: Calculate the test statistic. Using the formula for a two-sample t-test:
t = (x̄1 - x̄2) / √((s1^2 / n1) + (s2^2 / n2))
Where:
x̄1 = sample mean for Alabama
x̄2 = sample mean for Mississippi
s1 = population standard deviation for Alabama
s2 = population standard deviation for Mississippi
n1 = sample size for Alabama
n2 = sample size for Mississippi
Plugging in the values from the problem:
x̄1 = $199.65
x̄2 = $187.93
s1 = $32.48
s2 = $26.15
n1 = 70
n2 = 65
t = ($199.65 - $187.93) / √(($32.48^2 / 70) + ($26.15^2 / 65))
Step 4: Find the critical value(s). The critical value(s) depends on the significance level and the degrees of freedom. Since this is a two-sample t-test, the degrees of freedom can be calculated using the formula:
df = (s1^2 / n1 + s2^2 / n2)^2 / [(s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1)]
Plugging in the values:
df = (($32.48^2 / 70) + ($26.15^2 / 65))^2 / [(($32.48^2 / 70)^2) / (70 - 1) + (($26.15^2 / 65)^2) / (65 - 1)]
Step 5: Make a decision. If the calculated t-value is greater than the critical value, we reject the null hypothesis.
b. The p-value is given as 0.0439.
c. To determine if the null hypothesis would be rejected at a different significance level, we compare the p-value to the new significance level. If the p-value is less than the new significance level, we reject the null hypothesis.
Note: To complete the calculation, the specific values for the means, standard deviations, and sample sizes in the question need to be substituted into the formulas, and the critical value(s) need to be determined using a t-table or statistical software.
To test whether the two means are different, we can use a two-sample t-test. Let's break down the steps for each part of the question:
a. Testing the Two Means:
1. Set up the null and alternative hypotheses:
- Null hypothesis (Ho): The means are equal: μ1 = μ2
- Alternative hypothesis (Ha): The means are different: μ1 ≠ μ2
2. Select the level of significance (alpha) as 0.05 (or 5%).
3. Calculate the test statistic:
- Formula for the test statistic:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
- Substitute the given values into the formula:
t = (199.65 - 187.93) / sqrt((32.48^2 / 70) + (26.15^2 / 65))
4. Determine the degrees of freedom:
- Degrees of freedom (df) = (n1 - 1) + (n2 - 1) = 70 - 1 + 65 - 1 = 133
5. Find the critical value (t-critical):
- Since it is a two-tailed test at a 5% level of significance with 133 degrees of freedom, the critical value is approximately ±1.981.
6. Compare the calculated test statistic to the critical value:
- If the calculated test statistic falls within the critical region (i.e., outside the range ±1.981), we reject the null hypothesis.
- If the calculated test statistic falls outside the critical region, we fail to reject the null hypothesis.
b. P-value for this test:
1. If the p-value for the test is given as 0.0439, it means that the probability of observing the test statistic (or a more extreme value) under the null hypothesis is 0.0439.
2. Compare the p-value to the level of significance (alpha):
- If the p-value is less than the level of significance (0.05), we reject the null hypothesis.
- If the p-value is greater than or equal to the level of significance, we fail to reject the null hypothesis.
c. Rejection of Ho with a different level of significance:
1. If the level of significance used was 10% (or 0.10), we compare the p-value (0.0439) to this new significance level.
2. If the p-value is less than the level of significance (0.10), we reject the null hypothesis.
- In this case, since the p-value (0.0439) is less than 0.10, we would reject the null hypothesis at a 10% significance level.
However, we cannot determine the result for a 1% level of significance since the p-value (0.0439) is greater than 0.01.