Question

a sample have 10 students the average score in math is less than the average in...

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

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 = dbar +- tcritical sed

= 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.

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