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Can the samples be considered to have been drawn from the same population ?

The means of two random samples of size 9 and 7 are 196.42 and 198.82 respectively.The sum of the squares of the deviations from the means are 26.94 and 18.73 respectively. Can the samples be considered to have been drawn from the same population ?

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$\text{(i) The null hypothesis H$_0$ : μ$_1$= μ$_2$. Alternative hypothesis H$_a$ : μ$_1$≠ μ$_2$.} \ $ $\text{(ii) Calculation of test statistic: Since the sample size is small , we use t – distribution}$ $\text{S$_p$ = $\sqrt{ \frac{∑(X_i- \bar{X})^2 +∑(Y_i- \bar{Y})^2 }{(n_1+n_2-2)}}$ = $\sqrt{ \frac{(26.94+18.73)}{(9+7-2)}}$ = 1.81}$ $\text{S.E.= S$_p$ $\sqrt{(\frac{1}{n_1} + \frac{1}{n_2 })}$ = 1.81 $\sqrt{( \frac{1}{9}+ \frac{1}{7})}$ = 0.91 }$ $\text{t = $\frac{( \bar{X_1} - \bar{X_2})}{(S.E.)}$ = $\frac{(196.42-198.82)}{(0.91)}$ = - 2.64}$ $\text{(iii) Level of significance : α =0.05}$ $\text{(iv) Critical value : the value of t$_α$ at 5% level of significance for ν= 9+7 - 2 =14 degrees of freedom is 2.145}$ $\text{(v) Decision : since the calculated value of |t| =2.64 is greater than the table value t$_α$=2.145, the null hypothesis is rejected.}$ $\text{∴ The samples cannot be considered to have been drawn from the same population.}$

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