Hypothesis Testing

The Language of Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. The process involves formulating two competing hypotheses and using sample evidence to determine which is more plausible.

• Null Hypothesis (H0): The statement being tested, usually representing no effect or no difference.

• Alternative Hypothesis (Ha): The statement we consider if the evidence suggests H0 is unlikely.

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

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

• Significance Level (α): The probability of making a Type I error, commonly set at 0.05.

Example: Testing whether the average daily time spent on phone calls has increased from a known value.

Steps in Hypothesis Testing

1. State the null and alternative hypotheses.

2. Choose a significance level (α).

3. Check assumptions and conditions (e.g., normality, independence).

4. Calculate the test statistic using sample data.

5. Find the p-value or critical value.

6. Make a decision: reject or fail to reject H0.

7. State the conclusion in context.

Formula for Test Statistic (One-Sample z-test for Mean):

Confidence Intervals

Confidence Intervals for Means and Proportions

A confidence interval provides a range of plausible values for a population parameter, calculated from sample data. The confidence level (e.g., 95%) indicates the proportion of intervals that would contain the parameter in repeated samples.

• For Means: Used when estimating the population mean.

• For Proportions: Used when estimating the population proportion.

Formula for Confidence Interval for a Mean (when σ known):

Formula for Confidence Interval for a Proportion:

Example: Constructing a 95% confidence interval to judge whether the proportion of males with tattoos differs from females.

Two-Sample Tests

Comparing Two Means or Proportions

Two-sample tests are used to compare the means or proportions of two independent groups. The hypotheses typically test for equality or difference between the groups.

• Two-Sample t-Test: Compares means from two independent samples.

• Two-Proportion z-Test: Compares proportions from two independent samples.

Formula for Two-Sample t-Test Statistic:

Formula for Two-Proportion z-Test Statistic:

where is the pooled sample proportion.

Regression and Correlation

Simple Linear Regression

Regression analysis examines the relationship between two quantitative variables. Simple linear regression models the relationship as a straight line.

• Regression Equation:

• Slope (b1): The estimated change in y for a one-unit increase in x.

• Intercept (b0): The estimated value of y when x = 0.

• Correlation Coefficient (r): Measures the strength and direction of the linear relationship.

Example: Predicting the distance a baseball travels based on the speed at which it is hit.

Interpreting Regression Output

• Assess the fit of the model using residuals and R-squared.

• Check if the model is appropriate for prediction (e.g., avoid extrapolation).

Using the TI-84 Plus CE Calculator

Common Functions for Hypothesis Testing and Confidence Intervals

Note: Always check assumptions before using statistical tests or intervals.

Additional Topics Covered in Questions

• Identifying and interpreting p-values

• Determining when to use one-tailed vs. two-tailed tests

• Assessing independence in sampling methods

• Constructing and interpreting confidence intervals for means and proportions

• Evaluating statistical significance and practical significance

Example: Determining if the use of an online homework system improves exam scores using a two-sample t-test.

Additional info: These notes synthesize the exam review sheet and practice problems, expanding on the statistical concepts and procedures referenced in the questions. All formulas are standard for introductory statistics courses.