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 (σ) is known or unknown.

• Requirements:

  • Simple Random Sample (SRS)

  • Independent samples

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

• Confidence Interval (CI) for Mean:

  • If σ is known:

  • If σ is unknown:

• Hypothesis Test:

  • Null Hypothesis:

  • Alternative Hypothesis: (two-sided) or , (one-sided)

  • Test Statistic (known σ):

  • Test Statistic (unknown σ):

  • Decision: Compare test statistic to critical value or use p-value method.

• Type I and Type II Errors:

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

  • Type II Error (β): Failing to reject when it is false.

  • Power:

• Sample Size Calculation:

  • For desired margin of error :

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

Inferences for Two Population Means

Independent and Dependent Samples

Comparing two population means can involve independent samples (pooled or non-pooled) or dependent samples (matched pairs).

• Independent Samples:

  • Pooled t-test (assume equal variances):

    • Requirements: SRS, independent samples, populations are normal, equal variances

    • Pooled variance:

    • Test statistic:

  • Non-pooled t-test (unequal variances):

    • Requirements: SRS, independent samples, populations are normal

    • Test statistic:

    • Degrees of freedom:

• Dependent Samples (Matched Pairs):

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

  • Calculate differences

  • Mean and standard deviation of differences: ,

  • Test statistic:

  • Degrees of freedom:

• Example: Comparing test scores before and after a training program using matched pairs.

Inferences for Population Standard Deviations (Variances)

Testing and Confidence Intervals

Inference for variances involves using the chi-square and F-distributions.

• One Population Variance:

  • Requirements: SRS, normal population

  • Test statistic:

  • CI for variance:

• Two Population Variances:

  • Requirements: SRS, independent samples, normal populations

  • Test statistic:

  • CI for ratio of variances:

• Example: Testing if two machines produce products with equal variability.

Nonparametric Tests for Means

Wilcoxon Signed Rank Test and Mann-Whitney U Test

Nonparametric tests are used when data do not meet the assumptions of normality or when dealing with ordinal data.

• Wilcoxon Signed Rank Test (Paired Samples):

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

  • Calculate differences, rank absolute values, sum positive ranks

  • Test statistic: (sum of positive ranks)

  • Compare to critical values from Wilcoxon chart

• Mann-Whitney U Test (Independent Samples):

  • Requirements: SRS, independent samples, same shape distributions

  • Rank all values, sum ranks for each sample

  • Test statistic: or (sum of ranks for sample 1)

  • Compare to critical values from Mann-Whitney chart

• Example: Comparing median incomes between two groups using Mann-Whitney U test.

Summary Table: Inference Methods for Means and Variances

The following table summarizes the main inference methods for means and variances:

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