9.1 Two population means with unknown standard deviations -- rrc  (Page 5/24)

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Chapter review

Two population means from independent samples where the population standard deviations are not known

• Random Variable: ${\overline{X}}_1-{\overline{X}}_2$ = the difference of the sampling means
• Distribution: Student's t -distribution with degrees of freedom (variances not pooled)

Formula review

Standard error: SE = $\sqrt{\frac{{(s_1)}^2}{n_1}+\frac{{(s_2)}^2}{n_2}}$

Test statistic ( t -score): t = $\frac{({\overline{x}}_1-{\overline{x}}_2)-({\mu }_1-{\mu }_2)}{\sqrt{\frac{{(s_1)}^2}{n_1}+\frac{{(s_2)}^2}{n_2}}}$

Degrees of freedom:
$df= \frac{{\left(\frac{{(s_1)}^2}{n_1}+ \frac{{(s_2)}^2}{n_2}\right)}^2}{\left(\frac{1}{n_1-1}\right){\left(\frac{{(s_1)}^2}{n_1}\right)}^2+\left(\frac{1}{n_2-1}\right){\left(\frac{{(s_2)}^2}{n_2}\right)}^2}$

where:

s ₁ and s ₂ are the sample standard deviations, and n ₁ and n ₂ are the sample sizes.

${\overline{x}}_1$ and ${\overline{x}}_2$ are the sample means.

Cohen’s d is the measure of effect size:

$d=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_{pooled}}$
where $s_{pooled}=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}$

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for

1. independent group means, population standard deviations, and/or variances known
2. independent group means, population standard deviations, and/or variances unknown
3. matched or paired samples
4. single mean
5. two proportions
6. single proportion