Confidence Intervals

Formulas for Confidence Intervals

Confidence intervals are used to estimate population parameters based on sample statistics. The formulas vary depending on the parameter being estimated and the sample characteristics.

• Confidence Interval for Population Proportion: Used to estimate the proportion of a population with a certain characteristic.

  • Formula:

  • Where is the sample proportion, is the critical value from the standard normal distribution, and is the sample size.

• Confidence Interval for Population Mean (Known Standard Deviation): Used when the population standard deviation is known.

  • Formula:

  • Where is the sample mean, is the population standard deviation, and is the sample size.

• Confidence Interval for Population Mean (Unknown Standard Deviation): Used when is unknown; the sample standard deviation is used and the t-distribution is applied.

  • Formula:

  • Where is the critical value from the t-distribution with degrees of freedom.

• Confidence Interval for Difference of Two Proportions: Used to compare proportions from two independent samples.

  • Formula:

• Confidence Interval for Difference of Two Means (Independent Samples): Used to compare means from two independent samples.

  • Formula (Equal Variances):

  • Where is the pooled variance.

  • Formula (Unequal Variances):

• Confidence Interval for Paired Data (Dependent Samples): Used for matched pairs or repeated measures.

  • Formula:

  • Where is the mean difference, is the standard deviation of differences.

Key Table: Confidence Interval Formulas

The following table summarizes the main formulas for constructing confidence intervals for proportions and means, including both single and two-sample cases.

Hypothesis Testing

Test Statistics for Hypothesis Tests

Hypothesis testing is used to make inferences about population parameters. The choice of test statistic depends on the parameter, sample size, and whether population standard deviation is known.

• Proportion Test: Used to test hypotheses about population proportions.

  • Test Statistic:

  • Where is the hypothesized population proportion.

• Mean Test (Known ): Used when population standard deviation is known.

  • Test Statistic:

  • Where is the hypothesized population mean.

• Mean Test (Unknown ): Used when population standard deviation is unknown.

  • Test Statistic:

• Two Proportions Test: Used to compare two population proportions.

  • Test Statistic:

  • Where is the pooled proportion.

• Two Means Test (Independent Samples): Used to compare means from two independent samples.

  • Test Statistic (Equal Variances):

  • Test Statistic (Unequal Variances):

• Paired Data Test: Used for matched pairs or repeated measures.

  • Test Statistic:

Key Table: Hypothesis Test Statistics

The following table summarizes the main test statistics for hypothesis testing of proportions and means.

Critical Values and Using Statistical Software

Finding Critical Values

Critical values are used to determine the boundaries of confidence intervals and rejection regions in hypothesis tests. They are typically found using statistical tables or software.

• For Proportions: Use the standard normal () distribution.

• For Means: Use the distribution if is known, or the t-distribution if $\sigma$ is unknown.

• Software Tools: Statistical software can be used to find critical values for both left and right tail areas.

Example: Using Software to Find Critical Values

To find the critical value for a confidence interval or hypothesis test:

1. Open the statistical software and select the normal distribution.

2. Enter the desired confidence level or significance level ().

3. Choose left or right tail area as appropriate.

4. The software will display the critical value.

Visualizing Critical Regions

Critical regions are the areas in the tails of the distribution where the null hypothesis is rejected. The critical value marks the boundary of this region.

• Left Tail: The critical region is to the left of the critical value.

• Right Tail: The critical region is to the right of the critical value.

• Two-Tailed: The critical regions are in both tails, each with area .

Example: For a 95% confidence interval, , so .

Summary Table: Confidence Intervals and Hypothesis Tests

This table provides a quick reference for the formulas and test statistics used in confidence intervals and hypothesis testing for proportions and means.

Additional info: These notes cover material from Chapters 7, 8, and 9 of a typical statistics course, focusing on confidence intervals, hypothesis testing, and inferences from two samples.