AP Statistics 4.5-4.6: Comparing Two Population Means

Two-sample test for independent population means

A hypothesis test comparing the means of two distinct populations using two independent simple random samples. The random variable is the difference in sample means, $\bar X_1-\bar X_2$.

What conditions support inference for comparing two independent population means?

The samples should be independent simple random samples from distinct populations. If sample sizes are small, each population should be approximately normal; with sufficiently large samples, the population distributions need not be normal.

Why is the difference between two sample means standardized in a two-mean test?

A large observed difference may result from high variability rather than a real difference in population means. Dividing $\bar x_1-\bar x_2$ by its standard error accounts for sampling variation and produces a standardized test statistic.

Standard error for the difference between two independent means when population standard deviations are unknown

The estimated standard error is $\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$, where $s_1$ and $s_2$ are sample standard deviations and $n_1$ and $n_2$ are sample sizes.

Welch two-sample t-test

The Aspin-Welch t-test compares two independent population means when population standard deviations are unknown and may be unequal. It uses the separate sample variances rather than pooling them.

Test statistic for two independent means with unknown standard deviations

The Welch statistic is $t=\frac{(\bar x_1-\bar x_2)-(\mu_1-\mu_2)_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$, where $(\mu_1-\mu_2)_0$ is the hypothesized difference, usually $0$.

How are the degrees of freedom estimated for Welch's two-sample t-test?

Use the Welch-Satterthwaite approximation: $df=\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1}+\frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}$. A calculator or computer should normally perform this calculation, and the result need not be an integer.

Why are the variances not pooled in the unequal-standard-deviation two-sample t-test?

Pooling assumes the two populations have a common standard deviation. Welch's procedure allows the population standard deviations to differ, so it retains $s_1^2$ and $s_2^2$ separately in the standard error.

When is the Student's t approximation generally reliable for two independent means with unknown standard deviations?

For approximately normal populations, the approximation is especially reliable when both sample sizes are at least 5. When the combined sample size exceeds 30, a normal approximation to the t distribution may also be used in the treatment described.

How should the wording 'the two population means are the same' be translated into hypotheses?

Use $H_0:\mu_1=\mu_2$ or $H_0:\mu_1-\mu_2=0$ and, for a difference question, $H_a:\mu_1\ne\mu_2$. This is a two-tailed test.

How are hypotheses written when the research claim is that population 1 has a larger mean?

Use $H_0:\mu_1\le\mu_2$ (equivalently, $\mu_1-\mu_2\le0$) and $H_a:\mu_1>\mu_2$. This is a right-tailed test.

How are hypotheses written when the research claim is that population 1 has a smaller mean?

Use $H_0:\mu_1\ge\mu_2$ and $H_a:\mu_1<\mu_2$. This is a left-tailed test.

How does the alternative hypothesis determine the tail of a two-mean test?

A $>$ alternative produces a right-tailed test, a $<$ alternative produces a left-tailed test, and a $\ne$ alternative produces a two-tailed test. The null hypothesis contains equality or the boundary value.

How is a p-value used to make a decision in a two-population-mean hypothesis test?

Reject $H_0$ when $p\text{-value}<\alpha$; otherwise, do not reject $H_0$. A small p-value indicates that the observed difference, or a more extreme one in the direction of $H_a$, is unlikely if the null hypothesis is true.

What does 'do not reject the null hypothesis' mean in a two-mean test?

It means the sample provides insufficient evidence for the stated alternative at the chosen significance level. It does not prove that the two population means are equal.

How should a conclusion from a two-mean hypothesis test be stated?

State the significance level, whether evidence is sufficient, and the population claim in context. For example: 'At the $\alpha$ level, there is sufficient evidence to conclude that the mean of population 1 is greater than the mean of population 2.'

Why is the null difference often set to zero in a two-mean test?

A null hypothesis of no difference, $H_0:\mu_1=\mu_2$, is equivalent to $H_0:\mu_1-\mu_2=0$. The test statistic then compares the observed difference in sample means with zero after standardizing.

What is the effect of increasing either sample size on the standard error of a difference in means?

Increasing $n_1$ or $n_2$ decreases its contribution, because each variance term is divided by sample size. Therefore, larger samples generally make the estimated difference more precise.

Why can two noticeably different sample means occur by chance?

Sampling variability can produce different sample means even when the population means are equal. The likelihood of such a difference depends on both the size of the difference and the variability represented by the standard error.

A study compares two independent groups, has unknown and potentially unequal standard deviations, and yields $t=-3.14$ with $p=0.0054$. What is the appropriate conclusion at $\alpha=0.05$ for a two-tailed alternative?

Because $0.0054<0.05$, reject $H_0$. There is sufficient evidence that the two population means differ, with the negative statistic indicating that sample group 1's mean is lower than sample group 2's mean.

A one-sided test comparing two means produces $p=0.1928$ at $\alpha=0.01$. What decision should be made?

Do not reject $H_0$ because $0.1928>0.01$. The data do not provide sufficient evidence for the claimed directional difference at the 1% significance level.

What is the sampling distribution of $\bar X_1-\bar X_2$ for two independent samples?

Its mean is $\mu_1-\mu_2$. When the population standard deviations are unknown, its standard deviation is estimated by $\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$, and the standardized difference is modeled with a Student's $t$ distribution using Welch degrees of freedom.

What is the point estimate for the difference between two population means?

The point estimate of $\mu_1-\mu_2$ is the difference between the sample means, $\bar x_1-\bar x_2$.

What is the two-sample t confidence interval for $\mu_1-\mu_2$?

A two-sample t interval is $(\bar x_1-\bar x_2)\pm t^*\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$, where $t^*$ is based on the confidence level and Welch's estimated degrees of freedom.

What conditions are needed for a two-sample t interval for the difference between population means?

The two samples should be independent random samples from distinct populations. Each population should be approximately normal when sample sizes are small; for larger samples, the populations need not be normal, provided there are no extreme outliers or strong skewness.

How should a confidence interval for $\mu_1-\mu_2$ be interpreted?

A confidence interval gives a set of plausible values for the difference in the two population means. In context, we say we are confident at the stated level that the interval captures the true value of $\mu_1-\mu_2$.

How can a confidence interval be used to assess a claim about two population means?

Compare the claimed difference with the interval. If the claim's value is outside the interval, it is not plausible at the corresponding significance level. If the claim's value is inside the interval, the data are consistent with that claim.

What does it mean if a confidence interval for $\mu_1-\mu_2$ contains zero?

Zero is a plausible value for the difference, so the interval provides insufficient evidence of a difference between the population means at the corresponding significance level.

What does it mean if a confidence interval for $\mu_1-\mu_2$ is entirely positive or entirely negative?

An entirely positive interval supports $\mu_1>\mu_2$, while an entirely negative interval supports $\mu_1<\mu_2$. In either case, zero is not a plausible value for the difference at the corresponding confidence level.

How are a two-sample t confidence interval and a two-sided t test related?

A two-sided test of $H_0:\mu_1-\mu_2=0$ at significance level $\alpha$ agrees with a $100(1-\alpha)\%$ confidence interval: reject $H_0$ exactly when $0$ is outside the interval.