Confidence Intervals and Hypothesis Testing

Overview

This study guide covers key concepts from Chapters 9, 10, and 11 in college-level statistics, focusing on confidence intervals, hypothesis testing for means and proportions, and inference for two population parameters. These topics are essential for understanding how to estimate population values and test statistical claims using sample data.

Confidence Intervals

Critical Values and Confidence Levels

Confidence intervals provide a range of plausible values for a population parameter, based on sample data. The critical value depends on the desired confidence level and the type of parameter being estimated (mean or proportion).

• Critical value for mean (z or t): Use for large samples or known population standard deviation; use for small samples or unknown standard deviation.

• Critical value for proportion: Use from the standard normal distribution.

• Common confidence levels: 90%, 95%, 99% (higher confidence = wider interval).

Formula for confidence interval for a mean:

Formula for confidence interval for a proportion:

• Example: For a sample proportion and , a 95% confidence interval is .

Interpreting Confidence Intervals

• A 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals would contain the true population parameter.

• Interpretation should always reference the population and parameter being estimated.

• Example: "We are 95% confident that the true mean diameter of Douglas fir trees is between 192.03 cm and 147.83 cm."

Hypothesis Testing

Steps in Hypothesis Testing

Hypothesis testing is used to assess claims about population parameters.

1. State the hypotheses: Null hypothesis () and alternative hypothesis ().

2. Choose the significance level (): Common values are 0.05, 0.01.

3. Calculate the test statistic: Use z or t formulas depending on context.

4. Find the p-value: Probability of observing the test statistic under .

5. Make a decision: If p-value , reject ; otherwise, fail to reject .

Test statistic for proportion:

Test statistic for mean:

• Example: Testing if the proportion of adults with hypertension is higher in Hawaii than the national average.

Types of Errors

• Type I Error: Rejecting when it is true (false positive).

• Type II Error: Failing to reject when is true (false negative).

• Example: In a medical study, a Type I error might mean concluding a drug is effective when it is not.

Inference for Two Population Parameters

Comparing Two Means or Proportions

When comparing two groups, use confidence intervals and hypothesis tests for the difference between means or proportions.

• Independent samples: Use two-sample t-test or z-test.

• Dependent samples (paired): Use paired t-test.

Confidence interval for difference in proportions:

Confidence interval for difference in means (independent samples):

• Example: "We are 95% confident that alcohol reduces bone fracture rate by 0.95 to 5.17%."

Choosing the Correct Test

• Proportion vs. Mean: Use z-test for proportions, t-test for means.

• Paired vs. Independent: Paired tests are for matched or repeated measures; independent tests are for separate groups.

• Example: Comparing swimming times before and after using goggles (paired t-test).

Summary Table: Common Confidence Intervals and Tests

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and formulas are based on standard textbook approaches for these chapters.