When 100 students at UCF were given a calculus problem that required the use of
integration by parts, 95 were able to solve it. When 100 FAU students were given the
same problem, 68 were able to solve it.
What is the test statistic associated with the claim that the students at UCF have a
better understanding of calculus than those at FAU? (You must show work and/or
calculator commands to receive full credit for this exercise.)
To determine the test statistic associated with the claim that UCF students have a better understanding of calculus than FAU students, we can use the formula for a two-proportion z-test.
The formula for the test statistic is:
Test statistic (z) = (p1 - p2) / √((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2))
where:
- p1 and p2 are the proportions of UCF and FAU students who solved the calculus problem, respectively.
- n1 and n2 are the sample sizes of UCF and FAU students, respectively.
Given the information provided:
- For UCF students, p1 = 95/100 = 0.95, and n1 = 100.
- For FAU students, p2 = 68/100 = 0.68, and n2 = 100.
Now, substitute these values into the formula to calculate the test statistic:
Test statistic (z) = (0.95 - 0.68) / √((0.95 * (1 - 0.95) / 100) + (0.68 * (1 - 0.68) / 100))
Simplifying the expression inside the square root:
Test statistic (z) = (0.27) / √((0.95 * 0.05 / 100) + (0.68 * 0.32 / 100))
Calculating the values inside the square root:
Test statistic (z) = 0.27 / √(0.00475 + 0.02176)
Adding the values inside the square root:
Test statistic (z) = 0.27 / √0.02651
Take the square root of 0.02651:
Test statistic (z) = 0.27 / 0.16265
Dividing 0.27 by 0.16265:
Test statistic (z) ≈ 1.66
Therefore, the test statistic associated with the claim that the students at UCF have a better understanding of calculus than those at FAU is approximately 1.66.