STAT1170 6: Population Means

Comparing Population Means

Paired Data

• Hypothesis Tests address research questions about a target population.

• Sample, target population, and variable of interest must be identified.

• Example Research Questions include:

  • Average length of female green anacondas

  • Grocery prices in two cities

• Hypothesis Testing Process:

  • Null Hypothesis ($H<em>0$) and Alternative Hypothesis ($H</em>1$)

  • Check assumptions: Normal distribution for large samples (CLT) or approximate normality.

  • Calculate test statistic:

  • For $z$-test when $\sigma$ is known: $z = \frac{\bar{y} - \mu}{\frac{\sigma}{\sqrt{n}}}$

  • For $t$-test when $\sigma$ is unknown: $t = \frac{\bar{y} - \mu}{\frac{s}{\sqrt{n}}}$

  • Determine $p$-value and decide to reject or not reject $H_0$.

Paired t-Test

• Involves one sample with two measurements per subject (paired data).

• Examples include measuring weight before and after a program.

• Assume differences between pairs are normally distributed.

• Conduct test on the mean difference of the paired samples:

  • Null Hypothesis: $H<em>0: \mu</em>d = 0$

  • Test Statistic: $t = \frac{\bar{y}<em>d - \mu</em>d}{\frac{s<em>d}{\sqrt{n</em>d}}}$ where $n_d$ is the number of differences.

Two-Sample t-Test

• Compares means of two independent populations.

• Null Hypothesis: $H<em>0: \mu</em>1 = \mu<em>2$; Alternative: $H</em>1: \mu<em>1 \neq \mu</em>2$.

• Assumes equal variances ($\sigma<em>1 = \sigma</em>2$) for the test.

• Test Statistic calculated as:

  • $t = \frac{\bar{y}<em>1 - \bar{y}</em>2}{s<em>p \sqrt{\frac{1}{n</em>1} + \frac{1}{n_2}}}$

  • Where $s_p$ is the pooled standard deviation.

• Use descriptive statistics to summarize data for independent samples.

• Determine if evidence exists to reject $H_0$ based on $p$-value.

Conclusion

• Paired t-test used when observations are dependent; Two-sample t-test used for independent observations.

• The comparison of populations requires proper identification of null and alternative hypotheses, adequate sampling, and validation of assumptions for the respective tests.