# Test whether the claim that the two sample groups come from populations with the

Test whether the claim that the two sample groups come from populations with the same mean or not.
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irwchh
State the test hypotheses.
Let first population denote the treatment and the second population denote the placebo.
Null hypothesis:
${H}_{0}:{\mu }_{1}={\mu }_{2}$
Alternative hypothesis:
${H}_{1}:{\mu }_{1}\ne {\mu }_{2}$
Test statistic:
The test statistic formula is,
$t=\frac{\left({\stackrel{―}{x}}_{1}-{\stackrel{―}{x}}_{2}\right)-\left({\mu }_{1}-{\mu }_{2}\right)}{\sqrt{\frac{{s}_{p}^{2}}{{n}_{1}}+\frac{{s}_{p}^{2}}{{n}_{2}}}}$
Where ${s}_{p}^{2}=\frac{\left({n}_{1}-1\right){s}_{1}^{2}+\left({n}_{2}-1\right){s}_{2}^{2}}{\left({n}_{1}-1\right)+\left({n}_{2}-1\right)}$
With degrees of freedom $df={n}_{1}+{n}_{2}-2$
Let ${\stackrel{―}{x}}_{1}$ denotes the first sample mean, ${\stackrel{―}{s}}_{1}$ denotes the first sample standard deviation, ${n}_{1}$ denotes the first sample size, ${\stackrel{―}{x}}_{2}$ denotes the second sample mean, ${s}_{2}$ denotes the second sample standard deviation, ${n}_{2}$ denotes the second sample size, ${\mu }_{1}$ denotes the first sample population mean, and ${\mu }_{2}$ denotes the second sample population mean.
The pooled sample variance is,
${s}_{p}^{2}=\frac{\left({n}_{1}-1\right){s}_{1}^{2}+\left({n}_{2}-1\right){s}_{2}^{2}}{\left({n}_{1}-1\right)+\left({n}_{2}-1\right)}$
$=\frac{\left(22-1\right){\left(0.015\right)}^{2}+\left(22-1\right){\left(0.000\right)}^{2}}{\left(22-1\right)+\left(22-1\right)}$
$=\frac{0.004725+0}{42}$
$=0.0001125$
Substitute ${n}_{1}=22{\stackrel{―}{x}}_{1}=0.049,{s}_{p}^{2}=0.0001125,{n}_{2}=22$ and ${\stackrel{―}{x}}_{2}=0.000$ in the test statistic formula
$t=\frac{\left(0.049-0.000\right)}{\sqrt{\frac{0.0001125}{22}+\frac{0.0001125}{22}}}$
$=\frac{0.049}{0.0032}$