Question

Q3: Consider the Math and Writing test scores for twelve students:
Student 1: 540 M, 474 W; Student 2: 432 M, 380 W; Student 3: 528 M, 463 W; Student 4: 574 M, 612 W; Student 5: 448 M, 420 W; Student 6: 502 M, 526 W; Student 7: 480 M, 430 W; Student 8: 499 M, 459 W; Student 9: 610 M, 615 W; Student 10: 572 M, 541 W; Student 11: 390 M, 335 W; Student 12: 593 M, 613 W.
Make two assumptions, for convenience, that are not actually satisfied here:
Assumption 1: the sample sizes involved are large enough that our test statistics generally follow a normal distribution when the population variance is known;
Assumption 2: the sample sizes are large enough to imply that the relevant t-distribution and the standard normal distribution are identical.
a. Assume a significance level of 0.05. Formally state and conduct a two-sided hypothesis test about the difference in Math and Writing test scores using a test statistic ignoring the fact that the samples are matched.
b. Assume a significance level of 0.05. Formally state and conduct a two-sided hypothesis test about the difference in Math and Writing test scores using a test

statistic for matched samples.

Answer 1

data

Student	M	W
1	540	474
2	432	380
3	528	463
4	574	612
5	448	420
6	502	526
7	480	430
8	499	459
9	610	615
10	572	541
11	390	335
12	593	613

a)

Using Excel

data -> data analysis ->

t-Test: Two-Sample Assuming Unequal Variances
	M	W
Mean	514	489
Variance	4661.2727	8735.8182
Observations	12	12
Hypothesized Mean Difference	0
df	20
t Stat	0.7482
P(T<=t) one-tail	0.2315
t Critical one-tail	1.7247
P(T<=t) two-tail	0.4630
t Critical two-tail	2.0860

2-tailed p-value = 0.4630 > alpha

hence we fail to reject the null hypothesis

b)

t-Test: Paired Two Sample for Means
	M	W
Mean	514	489
Variance	4661.2727	8735.8182
Observations	12	12
Pearson Correlation	0.9422
Hypothesized Mean Difference	0
df	11
t Stat	2.3374
P(T<=t) one-tail	0.0197
t Critical one-tail	1.7959
P(T<=t) two-tail	0.0394
t Critical two-tail	2.2010

p-value = 0.0394 < 0.05

hence we reject the null hypothesis