Question

a sample have 10 students

the average score in math is less than the average in english

x1= test scores in math

x2= test scores in english

assume the scores follow normal distribution

let u1 be the population mean of math scored

let u2 be the population mean of english score

the score are given below

student. math. english

1. 18. 22

2. 22. 24

3. 19. 16

4. 23. 19

5. 20. 21

6. 18. 18

7. 16. 17

8. 17. 20

9. 21. 25.

10. 14. 15

A: are the observation independent?

B construct a 95% CI for the mean difference

C: test the hypothesis that whether the average score on math is less than average score in english at 5% level

Answer 1

A. No, the observation are not independent, as the samples are dependent.

B.

Student	Math	English	Difference
1	18	22	4
2	22	24	2
3	19	16	-3
4	23	19	-4
5	20	21	1
6	18	18	0
7	16	17	1
8	17	20	3
9	21	25	4
10	14	15	1
		Average	1.8
		Std.Dev.	1.6432

Here paired size n = 10

degree of freedom = 10 - 1

95% CI for mean difference = d_bar +- t_critical se_d

= 1.8 +- 2.26216 * 1.6432/sqrt(10)

= 1.8 +- 1.1755

=(0.6245, 2.9755)

(C) Here as the confidence interval doesn't consist the value of 0 so we can conclude that  the average score on math is less than average score in english at 5% level.