Let’s continue our analysis of liberals and conservatives, taking a look this time at differences in their educational attainment. We obtain the following information from the 2002 GSS – the average educational attainment for liberals is 13.90 years (SY = 3.27) and the average educational attainment for conservatives is 13.55 years (SY = 2.82). Data are based on 187 liberals and 227 conservative responses.
a. Test the research hypothesis that there is a difference in level of education between liberals and conservatives, set alpha at 0.01.
b. Would your decision have been different if alpha were set at 0.05?
I assume "SY" is your abbreviation for standard deviation (SD).
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
I'll let you do the calculations.
To test the research hypothesis that there is a difference in the level of education between liberals and conservatives, we can use a t-test.
a. Test at alpha = 0.01:
First, let's state our null and alternative hypotheses:
- Null hypothesis (H0): There is no difference in the level of education between liberals and conservatives.
- Alternative hypothesis (H1): There is a difference in the level of education between liberals and conservatives.
To test this, we will compare the means of the two groups using a t-test. The formula for a t-test is:
t = (mean1 - mean2) / sqrt((SY1^2 / n1) + (SY2^2 / n2))
where:
- mean1 and mean2 are the sample means of the two groups (13.90 and 13.55, respectively)
- SY1 and SY2 are the standard deviations of the two groups (3.27 and 2.82, respectively)
- n1 and n2 are the sample sizes of the two groups (187 liberals and 227 conservatives)
Now, let's calculate the t-value:
t = (13.90 - 13.55) / sqrt((3.27^2 / 187) + (2.82^2 / 227))
Using a statistical calculator or software, we find that the t-value is approximately 1.642.
Next, we need to compare this t-value to the critical t-value at alpha = 0.01 with (n1 + n2 - 2) degrees of freedom. Since the degrees of freedom in this case are (187 + 227 - 2) = 412, the critical t-value at alpha = 0.01 is approximately ±2.625.
Since the calculated t-value (1.642) is less than the critical t-value (±2.625), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that there is a significant difference in the level of education between liberals and conservatives at alpha = 0.01.
b. Test at alpha = 0.05:
If alpha were set at 0.05 instead, the critical t-value would be different. The new critical t-value would be approximately ±1.968 for the same degrees of freedom (412).
Since the calculated t-value (1.642) is still less than the new critical t-value (±1.968), we would still fail to reject the null hypothesis at alpha = 0.05. Therefore, our decision would not change even if alpha were set at 0.05.