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 or 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 or t > 5.841, since our test statistic isn't in this region, c is not a valid answer