The Professor's Response
A large sample confidence interval for the true average difference between quantitative GRE scores for engineering majors and physical science majors was created. The result was as follows:-45.09 < Î¼e - Î¼p < 95.09. Is there a significant difference? Which group had the larger sample mean?
For that specific problem, I don't know which interval is for engineering majors or physical science majors. I thought that 95.09 was for the physical science majors
See the professor's answer below.
There is a video for this exact problem: http://www.statsprofessor.com/video.php?chapterId=16&id=300#ptop
A large sample confidence interval for the true average difference between quantitative GRE scores for engineering majors and physical science majors was created. The result was as follows:-45.09 < μe - μp < 95.09. Is there a significant difference? Which group had the larger sample mean?
You asked which interval is for engineering majors. There is only one interval. An interval is a piece of the number line that includes all of the values between two end points. Our interval is from -45.09 to 95.09. I think you may have been wanting to ask me which end point value (-45.09 or 95.09) corresponds to which group (engineers or physical science majors). That question is not answerable because neither of the values corresponds to the two groups.
The interval is supposed to represent the values that could be equal to the true difference between population means. For example, zero is inside that interval, so it could be that the true difference between the mean for engineers and physical science majors is equal to zero. In other words, it could be that μe - μp = 0. It could also be true that μe - μp = 50, since 50 is inside the interval. If it were 50, that would imply that the mean for engineers (μe) is greater than the mean for physical science majors (μp) by 50 points. It could also be true that μe - μp = -30, since -30 is inside the interval. If it were -30, that would imply that the mean for engineers (μe) is less than the mean for physical science majors (μp) by 30 points.
Since all of these statements are possibly true and contradictory, the interval does not show a significant difference between the two groups. However, because there are more values on the positive side of the interval, it appears that the first sample mean was larger. The first sample mean is the sample mean for engineers, since the subtraction was done in the order μe - μp.
This can be seen analytically.
xbar-e - xbar-p - E = -45.09
xbar-e - xbar-p + E = 95.09
If we add the two equations, we get 2(xbar-e - xbar-p) = 50. Using basic Algebra, we divide both side by two to get xbar-e - xbar-p = 25.
Since the 25 is positive, we can tell that xbar for the engineers was 25 points higher than the xbar for the physical science majors. This is not a significant difference though because when the margin of error was added and subtracted, the interval went from a negative number to a positive number. This always indicates there is not a significant difference, since zero is inside the interval when this happens. That allows for the possibility that there is zero difference between the two groups.
Hope that answers your question,