Question

1. Mr. Canu and Emm administered their researcher to measure the mathematical ability of male students on some solid figures. They randomly took 10 males of Science High. Their scores in each solid figures are shown below. Use 0.05 level of significance
Student
1
2
3
4
5
6
7
8
9
10

Cube
10
7
5
5
4
3
7
3
5
8

Cylinder
5
7
7
6
6
5
6
4
5
8

Answer 1

Use 0.05 level of significance to determine whether the average mathematical ability of male students on cube is equal to the average mathematical ability of male students on cylinder.

First we will prove that population variances are equal.

$H_0: \sigma_x^2=\sigma_y^2,\ H_1: \sigma_x^2\neq\sigma_y^2\\ F=\frac{S_b^2}{S_s^2}\text{ where } S_b^2\text{ and } S_s^2 \text{ are bigger and smaller sample}\\ \text{variances respectively}.\\ F_{obs}\approx \frac{5.12}{1.43}\approx 3.58\\ k_1=k_2=n-1=10-1=9\\ F_{right\ cr.}=F_{cr.}(\alpha/2;k_1;k_2)\approx 4.03.\\ F_{obs.}<F_{right\ cr.}. \text{ So we accept }H_0.\\ H_0: \overline{x_p}=\overline{y_p},\ H_1: \overline{x_p}\neq\overline{y_p}\\ T=\frac{\overline{X}-\overline{Y}}{\sqrt{(n-1)S_x^2+(m-1)S_y^2}}\sqrt{\frac{nm(n+m-2)}{n+m}}\\ t_{obs}\approx -0.12\\ k=n+m-2=18\\ t_{cr.}=t_{cr.}(\alpha;k)\approx 2.1\\ (-\infty, -2.1)\cup (2.1, \infty)\text{ --- critical region}.\\ t_{obs} \text{ does not fall into the critical region}.\\ \text{We accept }H_0.\\ \text{There is no evidence that the average mathematical}\\ \text{ability of male students on cube is not equal to the}\\ \text{average mathematical ability of male students on cylinder}.$