Sample 1: 19 17 15 21 16 18 16 14

Sample 2: 15 14 15 19 15 18 16

Is the difference between sample means significant.

Mumbai University > Mechanical Engineering > Sem 4 > Applied Mathematics IV

Marks: 6M

Year: Dec 2014

$\begin{array}{l} {n_{1} =8\, \, and\, \, n_{2} =7\left(\lt30\, \, \, it\, \, \, is\, \, small\, \, sample\right)} \\ {Step\, \, 1\, \, :-} \\ {Null\, Hypothesis\, \, \left(H_{0} \right)=\mu _{1} =\mu _{2} \left(i.e.\, Different\, \, Between\, \, Simple\, \, \, is\, \, \, not\, \, {\rm Significantly}\, \right)\, } \\ {Alternative\, \, Hypothesis\, \, \left(H_{0} \right)=\mu _{1} \lt\mu _{2} \left(i.e.\, Different\, \, Between\, \, Simple\, \, \, is\, \, \, \, {\rm Significantly}\, \right)\left(Two\, \, tailed\, \, test\right)} \\ {Step\, \, 2:-} \\ {LOS\, =5\% \, \, \left(Two\, \, tailed\, \, \, test\right)} \\ {Degree\, \, of\, \, Freedom\, \, =n_{1} +n_{2} -2=8+7-2=13} \\ {{\mkern1mu\raise1pt\hbox{.}\mkern1mu\raise4pt\hbox{.}\mkern1mu\raise1pt\hbox{.}\mkern1mu} \, \, Critical\, \, Value\, \left(t_{\alpha } \right)=2.160} \end{array}$

Step 3 :-

$\overline{x_1}=\frac{\sum{x_{1t}}}{n_1}=\frac{136}{8}=17\ ;\overline{x_2}=\frac{\sum{x_{2t}}}{n_2}=\frac{112}{7}=16Since\ Sample\ is\ small,\ sp=\sqrt{\frac{\sum{{\left(x_1-\overline{x_1}\right)}^2+\sum{{\left(x_{2t}-\overline{x_2}\right)}^2}}}{n_1+n_2-2}}=\sqrt{\frac{36+20}{8+7-2}}$

$=2.0755$$

$S.E.=sp\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\mathrm{=2.0755\ }\sqrt{\frac{1}{8}+\frac{1}{7}}=1.0742$

Step 4 :- Test Statistic

$t_{cal} =\frac{\overline{x_{1} }-\overline{x_{2} }}{S.E.} =\frac{17-16}{1.0742} =0.9309$

Step 5 :- Decision

Since |teal|$\lt$t${}_{\mathrm{\propto }}$,H${}_{0}$ is accepted $\mathrm{\therefore }$ Different between sample mean is not significant.