Statistics for Business: Hypothesis Testing, Chi-Square, ANOVA, and Experimental Design Study Guide

Hypothesis Testing and Chi-Square Tests

Testing Relationships Between Categorical Variables

Statistical tests such as the chi-square test of independence are used to determine whether there is a significant relationship between two categorical variables, such as gender and favorite sport.

• Null Hypothesis (H0): Assumes no relationship between the variables (e.g., favorite sport is independent of gender).

• Alternative Hypothesis (HA): Assumes a relationship exists (e.g., favorite sport is not independent of gender).

• P-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. If the p-value is less than the significance level (commonly 0.05), the null hypothesis is rejected.

• Test Statistic for Chi-Square: $\chi^2 = \sum \frac{(O - E)^2}{E}$ where $O$ is the observed frequency and $E$ is the expected frequency.

• Standardized Residuals: Used to identify which cells contribute most to the chi-square statistic. Extreme values indicate cells with large deviations from expected counts.

Example: In a survey, the relationship between gender and favorite sport was tested using a chi-square test. The p-value was less than 0.05, indicating a significant relationship.

Confidence Intervals and Proportions

Constructing Confidence Intervals

Confidence intervals estimate the range within which a population parameter lies, based on sample data.

• Formula for Confidence Interval for Mean Difference (Two-Sample): $CI = (\bar{x}_1 - \bar{x}_2) \pm z^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ where $\bar{x}_1, \bar{x}_2$ are sample means, $s_1^2, s_2^2$ are sample variances, $n_1, n_2$ are sample sizes, and $z^*$ is the critical value.

• Confidence Level: The probability that the interval contains the true parameter (e.g., 95%).

• Interpreting Confidence Intervals: If the interval does not contain zero (for mean differences), there is evidence of a significant difference.

Example: A confidence interval for the difference in mean weights produced by two machines was calculated as (0.0155, 0.424), with a 95% confidence level.

Analysis of Variance (ANOVA)

One-Way ANOVA

ANOVA is used to compare means across multiple groups to determine if at least one group mean is different.

• Mean Square Error (MSE): Measures the average of the squared differences between observed and predicted values. $MSE = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

• Equal Variance Assumption: ANOVA assumes that the variances of the populations being compared are equal. This can be checked by comparing sample variances.

• Test Statistic: $F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

Example: In a one-way ANOVA with four populations, if all population variances are 15, the MSE should be close to 225.

Experimental Design and Types of Studies

Types of Study Designs

Understanding the design of a study is crucial for interpreting results and choosing appropriate statistical tests.

• Survey: Collects data from subjects without manipulation.

• Matched Pairs Design: Subjects are paired based on certain characteristics, and each pair receives different treatments.

• Independent Samples Design: Different subjects are assigned to different groups.

Example: In a study comparing travel times by two modes of transportation, a matched pairs design was used, pairing students and comparing their travel times.

Statistical Distributions

Chi-Square Distribution

The chi-square distribution is commonly used in tests of independence and goodness-of-fit for categorical data.

• Properties:

  • Skewed to the right (not symmetrical).

  • All values are non-negative.

  • Critical value for goodness-of-fit: $\chi^2 > \chi^2_{\alpha, k-1}$, where $k$ is the number of categories.

Example: The chi-square distribution is used to test whether observed frequencies differ from expected frequencies in categorical data.

Comparing Two Means and Proportions

Tests for Two Samples

When comparing two groups, different tests are used depending on the data type and assumptions.

• Two-Sample t-Test: Used for comparing means when population variances are unknown and possibly unequal. $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Pooled Proportion Test: Used for comparing proportions between two groups. $\hat{p}_1 - \hat{p}_2 \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Paired t-Test: Used when samples are paired or matched.

Example: To determine if the mean time worked is less in unsuccessful companies than successful companies, a two-sample t-test for means assuming unequal variances is appropriate.

Summary Tables

Example: Sport Preference Data

The following table summarizes the sport preference data by gender:

Gender	Basketball	Football	Golf	Tennis	Total
Male	30	20	35	35	120
Female	15	25	5	20	65
Total	45	45	40	55	185

Main Purpose: This table is used to analyze the relationship between gender and favorite sport using chi-square tests.

Example: Standardized Residuals Table

Gender	Basketball	Football	Golf	Tennis
Male	0.91	0.45	0.84	1.525
Female	-1.37	-0.68	-1.27	-2.31

Main Purpose: Identifies which sport and gender combinations have the largest deviations from expected counts.

Example: Summary Data for Two Samples

	Sample 1	Sample 2
Average, $\bar{x}$	15	13
Variance, $s^2$	8	9
Sample Size, $n$	10	12

Main Purpose: Used to calculate pooled variance and test statistics for comparing means.

Key Terms and Concepts

• Significance Level ($\alpha$): The probability of rejecting the null hypothesis when it is true, commonly set at 0.05.

• Critical Value: The threshold value that the test statistic must exceed to reject the null hypothesis.

• Degrees of Freedom: The number of independent values in a calculation, important for determining critical values.

• Matched Pairs: Experimental design where subjects are paired to control for confounding variables.

• Independent Samples: Groups are unrelated and randomly assigned.

Additional info: These notes expand on the original exam questions by providing definitions, formulas, and context for each statistical concept, ensuring a self-contained study guide for exam preparation.