Exam 1 Practice Problems: Key Concepts in Inferential Statistics

Hypothesis Testing for Two Population Means

Comparing Means from Two Independent Samples

When comparing the means of two independent populations, we use hypothesis tests to determine if there is a statistically significant difference between them. This is commonly applied in experiments or observational studies where two groups are compared.

• Null Hypothesis (H0): The population means are equal ($\mu_1 = \mu_2$).

• Alternative Hypothesis (Ha): The population means are not equal ($\mu_1 \neq \mu_2$).

• Test Statistic: For large samples, use the z-test; for small samples, use the t-test:

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom: Calculated using the sample sizes and variances.

• Example: Comparing the average lifespan of trees in two different locations using sample data.

Location	n	Mean	Std. Dev.
1	45	8.2	1.1
2	45	7.6	1.4

Additional info: The test compares the difference in means and assesses statistical significance using the calculated t-value and p-value.

Confidence Intervals for Population Proportions

Estimating Proportions in a Population

A confidence interval provides a range of plausible values for a population proportion, based on sample data. It is commonly used in survey research and public health studies.

• Sample Proportion (p̂): $\hat{p} = \frac{x}{n}$ where x is the number of successes and n is the sample size.

• Confidence Interval Formula:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• z* is the critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95%).

• Example: Estimating the proportion of children who meet recommended daily servings of fruits and vegetables.

Chi-Square Test for Independence

Testing Association Between Categorical Variables

The chi-square test for independence is used to determine whether there is a significant association between two categorical variables in a contingency table.

• Null Hypothesis (H0): The variables are independent.

• Alternative Hypothesis (Ha): The variables are not independent.

• Test Statistic:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

• O = observed frequency, E = expected frequency

• Degrees of Freedom: (number of rows - 1) × (number of columns - 1)

• Example: Survey of adults on preferred social media platform by age group.

Platform	Under 40	40+
X (Twitter)	18	7
Facebook	10	21
Reddit	16	5
Stocks	6	7

Additional info: The test determines if age and platform preference are associated.

Chi-Square Goodness-of-Fit Test

Testing Distribution Fit for Categorical Data

The chi-square goodness-of-fit test is used to determine if observed frequencies match expected frequencies for a categorical variable, such as outcomes of a die roll.

• Null Hypothesis (H0): The observed distribution matches the expected distribution.

• Alternative Hypothesis (Ha): The observed distribution does not match the expected distribution.

• Test Statistic:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

• Example: Rolling an eight-sided die 80 times and comparing observed counts to expected counts (10 per outcome).

Outcome	1	2	3	4	5	6	7	8
Count	11	8	12	7	11	12	12	7

Additional info: The test assesses whether the die is fair (all outcomes equally likely).

Summary Table: Statistical Tests Covered

Test	Purpose	Key Formula
Two-Sample t-Test	Compare means of two independent groups	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
Confidence Interval for Proportion	Estimate population proportion	$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Chi-Square Test for Independence	Test association between categorical variables	$\chi^2 = \sum \frac{(O - E)^2}{E}$
Chi-Square Goodness-of-Fit	Test fit of observed to expected distribution	$\chi^2 = \sum \frac{(O - E)^2}{E}$