Do students tend to improve their SAT mathematics (SAT-M) score the second time they take the test? A random sample of four students who took the test twice received the following scores: Student 1 2 3 4 First Score 450 520 720 600 Second Score 440 600 720 630 Assuming that the change in SAT-M score (second score - first score) for the population of all students taking the test twice is Normally distributed with mean μ, are we convinced that retaking the test improves scores? What is the P-value for a test of H0: μ = μ0 versus Ha: μ ≠ μ0? c. less than 0.01 a. more than 0.75 b. more than 0.10

b

Step-by-step explanation:

We will conduct a hypothesis test for dependent samples since the same subjects were retested.

We first need to find the difference in scores

Student 1: 450 - 440 = 10

Student 2: 520 - 600 = - 80

Student 3: 720 - 720 = 0

Student 4: 600 - 630 = - 30

Now find the mean of the differences: d = (10 - 80 + 0 - 30) / 4 = - 100/4 = - 25

Now find the standard deviation, which is the square root of the variance:

[ (10 - (-25)) ² + (-80 - (-25)) ² + (0 - (-25)) ² + (-30 - (-25)) ²] / (4*3) =

[35² + (-55) ² + 25² + (-5) ²]/12 = 4900/12 = 408.3333

s = √ (4900/12) = 20.2073

Find the test statistic: t = (d - µ) / (s/√n)

t = (-25 - 0) / (20.2073/√4) = - 2.474

For a critical region of this two tailed test, for a 90% significance level, we get a t-value of: t 2.353, since our test statistic is in this region, b is a valid answer

For a critical region of this two tailed test, for a 99% significance level, we get a t-value of: t 5.841, since our test statistic isn't in this region, c is not a valid answer