Estimation and Hypothesis Testing

Vocabulary and Notation

This section introduces key terms and notation used in statistical inference, including estimation and hypothesis testing.

• Point Estimate: A single value used to estimate a population parameter (e.g., sample mean \( \bar{x} \) estimates population mean \( \mu \)).

• Confidence Interval: A range of values, derived from sample statistics, that is likely to contain the population parameter.

• Level of Confidence: The probability that the confidence interval contains the true parameter, usually expressed as \( (1-\alpha) \cdot 100\% \).

• Margin of Error: The maximum expected difference between the point estimate and the true parameter.

• Critical Value: The value that marks the boundary for the desired confidence level (e.g., \( z_{\alpha/2} \) or \( t_{\alpha/2} \)).

• Student’s t-Distribution: A probability distribution used when estimating population mean with unknown variance and small sample size.

• Bootstrapping: A resampling method for estimating the distribution of a statistic.

• Percentile Method Confidence Interval: Uses percentiles from bootstrap samples to construct confidence intervals.

• Hypothesis: A claim about a population parameter. Includes Null Hypothesis (H0) and Alternative Hypothesis (H1).

• Hypothesis Testing: Procedure to assess evidence against H0 in favor of H1.

• Type I Error: Rejecting H0 when it is true.

• Type II Error: Failing to reject H0 when H1 is true.

• P-value: Probability of observing data as extreme as the sample, assuming H0 is true.

• Statistical Significance: When the P-value is less than the significance level (\( \alpha \)), indicating evidence against H0.

• Practical Significance: Whether the result has real-world importance.

• Independent Samples: Samples with no relationship between observations.

• Dependent Samples (Matched Pairs): Samples where observations are paired or related.

• Robust Test: A test that remains valid under violations of assumptions.

• Randomization Test: Uses random resampling to assess significance.

One Sample Confidence Intervals

Confidence Interval for One Sample Proportion

Used to estimate the population proportion based on a sample.

• Formula:

  • Lower bound:

  • Upper bound:

• Assumptions:

  • Sample obtained by simple random sampling or randomized experiment.

  • Sampled values are independent; sample size is less than 5% of population.

• Example: If , , and for 95% confidence, calculate bounds.

t Confidence Interval for Mean

Used to estimate the population mean when the population standard deviation is unknown.

• Formula:

  • Lower bound:

  • Upper bound:

• Assumptions:

  • Sample obtained by simple random sampling or randomized experiment.

  • No outliers; population is normally distributed or .

  • Sampled values are independent.

• Example: , , , for 95% confidence.

One Sample Hypothesis Tests

z Test for One Sample Proportion

Tests whether the sample proportion differs from a hypothesized value.

• Hypotheses:

  • Two-tailed: ,

  • Left-tailed: ,

  • Right-tailed: ,

• Test Statistic:

• Assumptions: Same as confidence interval for proportion.

t Test for Mean

Tests whether the sample mean differs from a hypothesized value.

• Hypotheses:

  • Two-tailed: ,

  • Left-tailed: ,

  • Right-tailed: ,

• Test Statistic:

• Degrees of Freedom:

• Assumptions: Same as t confidence interval for mean.

Two Sample Hypothesis Tests

Two Sample z Test for Proportions

Tests whether two population proportions are equal.

• Hypotheses:

  • Two-tailed: ,

  • Left-tailed: ,

  • Right-tailed: ,

• Test Statistic: where

• Assumptions: Samples are independent, random, and for each sample.

Two Sample t Test for Dependent Means (Matched Pairs)

Tests whether the mean difference in paired data is zero.

• Hypotheses:

  • Two-tailed: ,

  • Left-tailed: ,

  • Right-tailed: ,

• Test Statistic:

• Degrees of Freedom:

• Assumptions: Same as one sample mean, applied to differences.

Two Sample t Test for Independent Means (Unequal Variances)

Tests whether two population means are equal, assuming unequal variances.

• Hypotheses:

  • Two-tailed: ,

  • Left-tailed: ,

  • Right-tailed: ,

• Test Statistic:

• Degrees of Freedom:

• Assumptions: Samples are independent, random, populations are normal or sample sizes are large.

Two Sample Confidence Intervals

Confidence Interval for Difference Between Two Proportions

Estimates the difference between two population proportions.

• Formula:

  • Lower bound:

  • Upper bound:

• Assumptions: Samples are independent, random, and , .

Confidence Interval for Mean of Differences (Paired Data)

Estimates the mean difference in paired data.

• Formula:

  • Lower bound:

  • Upper bound:

• Degrees of Freedom:

• Assumptions: Same as one sample mean, applied to differences.

Confidence Interval for Difference Between Two Independent Means (Unequal Variances)

Estimates the difference between two population means.

• Formula:

  • Lower bound:

  • Upper bound:

• Degrees of Freedom:

• Assumptions: Samples are independent, random, populations are normal or sample sizes are large.

Sample Size Calculations

Sample Size Needed for Proportions

Calculates the minimum sample size required to estimate a proportion with a specified margin of error.

• When prior estimate is available:

• When no prior estimate:

• Where is the desired margin of error (as a decimal).

Sample Size Needed for Means

Calculates the minimum sample size required to estimate a mean with a specified margin of error.

• Formula:

• Where is the desired margin of error.

Summary Table: Hypothesis Tests and Confidence Intervals

Additional info: Bootstrapping and randomization tests are modern alternatives for inference, especially when assumptions are questionable. StatCrunch and similar software can be used for calculations and simulations.