Inferences for One Population Mean

Confidence Intervals and Hypothesis Tests

Statistical inference for a single population mean involves estimating the mean and testing hypotheses about its value. The methods depend on whether the population standard deviation is known or unknown.

• Confidence Interval (CI): An interval estimate for the population mean, calculated using sample data.

• Hypothesis Test: A procedure to decide whether a sample provides enough evidence to reject a stated hypothesis about the population mean.

• Requirements:

  • Simple Random Sample (SRS)

  • Independent samples

  • Population is normally distributed or sample size is large (Central Limit Theorem applies)

• Formulas:

  • When population standard deviation () is known:

  • When is unknown:

  • Test statistic (known ):

  • Test statistic (unknown ):

• Example: Suppose , , , . The CI is .

Inferences for Two Population Means

Independent Samples: Pooled and Non-Pooled t-Tests

Comparing two population means can be done using independent samples (pooled or non-pooled) or dependent samples (paired data).

• Pooled t-Test: Used when variances are assumed equal.

  • Requirements: SRS, independent samples, populations are normally distributed, equal variances.

  • Test statistic:

  • Pooled variance:

• Non-Pooled t-Test: Used when variances are not assumed equal.

  • Requirements: SRS, independent samples, populations are normally distributed.

  • Test statistic:

• Dependent Samples (Matched Pairs):

  • Requirements: SRS, dependent samples, differences are normally distributed.

  • Test statistic:

  • is the mean of the differences, is the standard deviation of the differences, is the number of pairs.

• Example: Comparing test scores before and after a training program using paired t-test.

Inferences for Population Standard Deviations and Variances

F-Test for Two Variances

To compare the variances of two populations, the F-test is used. This test is sensitive to the assumption of normality.

• Requirements: SRS, independent samples, populations are normally distributed.

• Hypotheses:

  • (or , depending on the alternative)

• Test statistic:

• Decision: Compare F to critical values from the F-distribution table.

• Example: Testing if the variance in test scores differs between two classes.

Nonparametric Tests for Means

Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

Used to compare two independent samples when the normality assumption is not met.

• Requirements: SRS, independent samples, same shape distributions.

• Procedure:

  • Rank all observations from both samples together.

  • Sum the ranks for each sample.

  • Calculate U statistic and compare to critical values.

• Example: Comparing median incomes between two groups.

Wilcoxon's Signed Rank Test (Paired Data)

Used for paired samples when the differences are not normally distributed.

• Requirements: SRS, dependent samples, symmetric distribution of differences.

• Procedure:

  • Calculate differences between pairs.

  • Rank the absolute differences, assign signs.

  • Sum positive and negative ranks.

  • Compare test statistic to critical values.

• Example: Comparing before and after measurements in a clinical trial.

Errors and Power in Hypothesis Testing

Type I and Type II Errors, Power

Understanding errors and power is crucial for interpreting hypothesis tests.

• Type I Error (): Rejecting the null hypothesis when it is true.

• Type II Error (): Failing to reject the null hypothesis when it is false.

• Power: The probability of correctly rejecting a false null hypothesis ().

• Example: If , there is a 5% chance of a Type I error.

Summary Table: Tests for Means and Variances

The following table summarizes the main tests discussed:

Additional info: Some formulas and procedures were inferred and expanded for clarity and completeness based on standard introductory statistics curriculum.