comparing two means and two variances

Point Estimator for the Difference Between Two Means

$$widehat{mu_{1}-mu_{2}} :=widehat{mu}_{1}-widehat{mu}_{2}=overline{X}_{1}-overline{X}_{2}$$

Distribution of $overline{X}_{1}-overline{X}_{2}$

2.9.1. Theorem. Let $overline{X}_{1}$ and $overline{X}_{2}$ be the sample means based on independent random samples of sizes $n_{1}$ and $n_{2}$ drawn from normal distributions with means $mu_{1}$ and $mu_{2}$ and variance $sigma_{1}^{2}$ and $sigma_{2}^{2},$ respectively. Then $overline{X}_{1}-overline{X}_{2}$ is normal with mean $mu_{1}-mu_{2}$ and variance $sigma_{1}^{2} / n_{1}+sigma_{2}^{2} / n_{2}​$. i.e,

$$frac{overline{X}_{1}-overline{X}_{2}-left(mu_{1}-mu_{2}right)}{sqrt{sigma_{1}^{2} / n_{1}+sigma_{2}^{2} / n_{2}}}$$

is a standard normal random variable.

As usual, the Central Limit theorem allows us to apply this result even to non-normal populations if we have large sample sizes.

Let $X_{gamma_{1}}^{2}$ and $X_{gamma_{2}}^{2}$ be independent chi-squared random variables with $gamma_{1}$ and $gamma_{2}$ degrees of freedom:

$$F_{gamma_{1}, gamma_{2}}=frac{X_{gamma_{1}}^{2} / gamma_{1}}{X_{gamma_{2}}^{2} / gamma_{2}}$$

is said to follorw an F-distribution。

Remark:

$$Pleft[F_{gamma_{1}, gamma_{2}}<xright]=Pleft[1 / F_{gamma_{1}, gamma_{2}}>1 / xright]=1-Pleft[F_{gamma_{2}, gamma_{1}}<1 / xright]$$ $$f_{1-alpha, gamma_{1}, gamma_{2}}=frac{1}{f_{alpha, gamma_{2}, gamma_{1}}}$$

If $sigma_{1}^{2}=sigma_{2}^{2},$ then the statistic

$$frac{S_1^2}{S_2^2}$$

follows an $F$ -distribution with $n_{1}-1$ and $n_{2}-1$ degrees of freedom.

The F -Test

$$F_{n_{1}-1, n_{2}-1}=frac{S_{1}^{2}}{S_{2}^{2}}$$

We reject at significance level $alpha $:

$$begin{array}{l}{H_{0} : sigma_{1} leq sigma_{2} text { if } frac{S_{1}^{2}}{S_{2}^{2}}>f_{alpha, n_{1}-1, n_{2}-1}} \ {H_{0} : sigma_{1} geq sigma_{2} text { if } frac{S_{2}^{2}}{S_{1}^{2}}>f_{alpha, n_{2}-1, n_{1}-1}} \ {H_{0} : sigma_{1}=sigma_{2} text { if } frac{S_{1}^{2}}{S_{2}^{2}}>f_{alpha / 2, n_{1}-1, n_{2}-1} text { or } frac{S_{2}^{2}}{S_{1}^{2}}>f_{alpha / 2, n_{2}-1, n_{1}-1}}end{array}$$

Pooled Variance

$$S_{p}^{2}=frac{left(n_{1}-1right) S_{1}^{2}+left(n_{2}-1right) S_{2}^{2}}{n_{1}+n_{2}-2}$$

Pooled T-Ｔest

$$begin{aligned} T_{n_{1}+n_{2}-2} &=frac{Z}{sqrt{X_{n_{1}+n_{2}-2}^{2} /left(n_{1}+n_{2}-2right)}} \ &=frac{left(overline{X}_{1}-overline{X}_{2}right)-left(mu_{1}-mu_{2}right)}{sqrt{S_{p}^{2}left(1 / n_{1}+1 / n_{2}right)}} end{aligned}$$

follows a T-distribution with $n_1 + n_2 -2$ degrees of freedom.

We reject at significance level $alpha $:

$$begin{array}{l}{H_{0} : mu_{1}-mu_{2}=left(mu_{1}-mu_{2}right)_{0} text { if }left|T_{n_{1}+n_{2}-2}right|>t_{alpha / 2, n_{1}+n_{2}-2}} \ {H_{0} : mu_{1}-mu_{2} leqleft(mu_{1}-mu_{2}right)_{0} text { if } T_{n_{1}+n_{2}-2}>t_{alpha, n_{1}+n_{2}-2}} \ {H_{0} : mu_{1}-mu_{2} geqleft(mu_{1}-mu_{2}right)_{0} text { if } T_{n_{1}+n_{2}-2}<-t_{alpha, n_{1}+n_{2}-2}}end{array}$$

Confidence Interval for the Difference of Means

$$left(overline{X}_{1}-overline{X}_{2}right) pm t_{alpha / 2, n_{1}+n_{2}-2} sqrt{S_{p}^{2}left(1 / n_{1}+1 / n_{2}right)}$$

Pooled test for equality of means

$$T_{gamma}=frac{left(overline{X}_{1}-overline{X}_{2}right)-left(mu_{1}-mu_{2}right)_{0}}{sqrt{S_{1}^{2} / n_{1}+S_{2}^{2} / n_{2}}}$$

We reject at significance level $alpha $:

$$begin{array}{l}{H_{0} : mu_{1}-mu_{2}=left(mu_{1}-mu_{2}right)_{0} text { if }left|T_{gamma}right|>t_{alpha / 2, gamma}} \ {H_{0} : mu_{1}-mu_{2} leqleft(mu_{1}-mu_{2}right)_{0} text { if } T_{gamma}>t_{alpha, gamma}} \ {H_{0} : mu_{1}-mu_{2} geqleft(mu_{1}-mu_{2}right)_{0} text { if } T_{gamma}<-t_{alpha, gamma}}end{array}$$

Define $D = X-Y$, whose mean will be

$$mu_{D}=mathrm{E}[D]=mathrm{E}[X-Y]=mathrm{E}[X]-mathrm{E}[Y]=mu_{X}-mu_Y$$

Paired T-Test

$$T=frac{overline{D}}{s_{D} / sqrt{n}}$$

Confidence bounds on $mu_X-mu_Y$ for paired data

$$overline{D} pm t_{alpha/2} S_d / sqrt{n}$$

The Wilcoxon Rank-Sum Test

$$H_{0} : P[X>Y]=frac{1}{2} quad text { or } quad H_{0} : P[X>Y] leq frac{1}{2}$$

m: smaller szmple size between $X$ and $Y$.

for large values of $m$, $W_m$ is approximately normally distributed with

$$mathrm{E}left[W_{m}right]=frac{m(m+n+1)}{2}, quad operatorname{Var} W_{m}=frac{m n(m+n+1)}{12}$$