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

Question

Test whether the claim that the two sample groups come from populations with the same mean or not.

Answer 1

State the test hypotheses.
Let first population denote the treatment and the second population denote the placebo.
Null hypothesis:

H_{0} : μ_{1} = μ_{2}

Alternative hypothesis:

H_{1} : μ_{1} \neq μ_{2}

Test statistic:
The test statistic formula is,

t = \frac{({\overset{―}{x}}_{1} - {\overset{―}{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{s_{p}^{2}}{n_{1}} + \frac{s_{p}^{2}}{n_{2}}}}

Where

s_{p}^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{(n_{1} - 1) + (n_{2} - 1)}

With degrees of freedom

d f = n_{1} + n_{2} - 2

Let

{\overset{―}{x}}_{1}

denotes the first sample mean,

{\overset{―}{s}}_{1}

denotes the first sample standard deviation,

n_{1}

denotes the first sample size,

{\overset{―}{x}}_{2}

denotes the second sample mean,

s_{2}

denotes the second sample standard deviation,

n_{2}

denotes the second sample size,

μ_{1}

denotes the first sample population mean, and

μ_{2}

denotes the second sample population mean.
The pooled sample variance is,

s_{p}^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{(n_{1} - 1) + (n_{2} - 1)}

= \frac{(22 - 1) {(0.015)}^{2} + (22 - 1) {(0.000)}^{2}}{(22 - 1) + (22 - 1)}

= \frac{0.004725 + 0}{42}

= 0.0001125

Substitute

n_{1} = 22 {\overset{―}{x}}_{1} = 0.049, s_{p}^{2} = 0.0001125, n_{2} = 22

and

{\overset{―}{x}}_{2} = 0.000

in the test statistic formula

t = \frac{(0.049 - 0.000)}{\sqrt{\frac{0.0001125}{22} + \frac{0.0001125}{22}}}

= \frac{0.049}{0.0032}

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