Answer to Question #173586 in Statistics and Probability for ANJU JAYACHANDRAN

Question

3. a) Two samples of 9 and 8 sizes give the sum of squares of deviations from respective

means equal to 160 and 91 inches squares. Test whether these samples have been

drawn from same normal population or not? (Use 05.0 α = )

Answer 1

Step 1 :-

Null Hypothesis: $(H_o)$ \Rightarrow\sigma_x^2=\sigma_y^2 \ (i.e. Samples are drawn from the same population) 

Alternate Hypothesis: $(H_{1})\Rightarrow \sigma_x^2\neq\sigma_y^2$ (i.e. Samples are not drawn from the same population)

Step 2 :-

LOS=5% (Two tailed test) 

Degree of freedom = $n_1+n_2-2=9+8-2=15$

Step 2 : Data

m = 9, n = 8

$\sum(x_i-\bar x)^2=160\ \ \ \sum(y_i-\bar y)^2=91$

Step 3 : Level of significance

α = 0.05

Step 4 : Test Statistic

$F=\dfrac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2}=\dfrac{S_1^2}{S_2^2},\ Under\ H_o$

Step 5 : Calculation

$s_x^2=\dfrac{1}{m-1}\sum(x_i-\bar x)^2=\dfrac{160}{8}=20\\ s_y^2=\dfrac{1}{n-1}\sum(y_i-\bar y)^2=\dfrac{91}{7}=13$

$F_o=\dfrac{s_x^2}{s_y^2}=\dfrac{20}{13}=1.54$

Step 6 : Critical values

Since H₁ is a two-sided alternative hypothesis the corresponding critical values are:

Critical value for 15 degree of freedoms= $\pm 2.1314$

Step 7 : Decision

Since f _{(8, 7),0.95} = -2.1314 < F₀ = 1.54 < f _{(8, 7),0.05} = 2.1314, the null hypothesis is not rejected and we conclude that Samples are drawn from the same population