Question

ten students were intensive coaching in statistics. the score obtained in first and fifth test are given below. does the score from first test and fifth test show an improvement.test at 5% level of significance. given ftab 1.833
1st test 7 8 4 9 6 5 8 9 4 7
5th test 6 6 6 9 7 4 8 7 3 8

Answer 1

Let's find means of 1st and 5th test:

$\overline{x_1}=(7+8+4+9+6+5+8+9+4+7)/10=6.7$

$\overline{x_5}=(6+6+6+9+7+4+8+7+3+8)/10=6.4$

To test whether is any significant difference between two means, we need to apply t-test.

$\vert t \vert=(\overline{x_1}-\overline{x_5})/(s*\sqrt{1/n_1+1/n_5})$

$n_1=n_5=n=10$

$s_1=\sqrt{\frac {\sum(x-\overline{x_1})^2} {n-1}}=\sqrt{\frac {\sum(x-\overline{x_1})^2} {n-1}}=\sqrt{\frac {32.1} {9}}=1.889$

$s_5=\sqrt{\frac {\sum(x-\overline{x_5})^2} {n-1}}=\sqrt{\frac {\sum(x-\overline{x_5})^2} {n-1}}=\sqrt{\frac {30.4} {9}}=1.838$

$s=\sqrt{(n*s_1^2+n*s_5^2)/(2n -2)}=\sqrt{(10*1.889^2+10*1.838^2)/(2*10 -2)}=\sqrt{(35.68+33.78)/(18)}=\sqrt{3.86}=1.964$

$\vert t \vert=(6.7-6.4)/(1.964*\sqrt{1/10+1/10})=0.342$

As $\vert t \vert$ is lower than tabulated value (1.734) there is no significant difference between scores of the first and the fifth test.

Answer: no significant improvement