Alta Chapter 9 - Hypothesis Testing

Hypothesis Testing Overview

• Null Hypothesis (H0): Represents a statement of no effect or no difference; often reflects a status quo.
• Alternative Hypothesis (Ha): Represents what the researcher aims to prove; suggests a difference or effect.
• Significance Level (α): Defines the probability of rejecting the null hypothesis when it is true, often set at 0.05 or 0.10.

Hypothesis Testing Procedure

• Identify the null and alternative hypotheses.
• Determine the significance level (α) for the test.
• Collect data to compute the test statistic.
• Compare the test statistic against a critical value to decide whether to reject H0.

Test Statistics

• Z-Score Calculation:

  • Used when population standard deviation is known.
  • Formula: ( z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} )
  • Example: Olivia's test yields a z-score of 3.87 when examining golf drive distances.

• Sample Mean and Standard Deviation:

  • Critical for calculating the test statistic.
  • Understanding sample sizes and their impacts on the results (larger samples lead to more reliable estimates).

Common Scenarios

• Different Claims:
  • A golfer (Olivia) hypothesizes her drive distance is greater than an established average; specific statistical calculations were completed for her case using standard deviation and sample calculations.
• Baker's Claim:
  • William's hypothesis about bread height's average was tested; obtained a z-score revealing whether to accept or reject his claims based on a threshold.

Example Outcomes

• Successful and Unsuccessful Tests:
  • Results may lead to rejecting or failing to reject H0 based on whether the calculated test statistics fell within the rejection region defined by the critical values.
  • Example of a failed rejection in the context of waitstaff tips indicating that average tips do not differ significantly from established norms.

Special Cases for Proportions

• Situations may arise requiring hypothesis testing for proportions, analyzed through different z-scores.
• Formulation includes determining sample proportions, successful events, and failure rates.

Selecting the Right Test

• T-Tests vs. Z-Tests:
  • T-tests are appropriate when the population standard deviation is unknown and the sample size is small.
  • Require knowledge of the sample mean, sample size, and sample standard deviation.

Two-Tailed vs. One-Tailed Tests

• Understanding when to apply left-tail, right-tail, or two-tailed tests based on the nature of the hypotheses.
• Example from a linguist’s study about study time for languages showcases the importance of tail direction.

Conclusions from Testing

• Deciding whether to reject or accept the null hypothesis based on calculated p-values and corresponding significance levels plays a critical role in hypothesis testing outcomes.

Key Calculations

• Results include detailed calculations for various scenarios such as:
  • z-score examples for Olivia (3.87), baker (some higher z-scores) leading to evidence rejection or acceptance of claims.
  • Various calculated test statistics rounded to two decimal places for clarity and communication of results.

Summary

• Hypothesis testing is a systematic approach to validate claims using statistical evidence, critical for decision-making in research and practical applications across various fields.### Hypothesis Testing Overview
• Null hypothesis (H0) posits that the population mean is a specific value (e.g., $250,000 for Rebecca).
• Alternative hypothesis (Ha) suggests that the population mean is greater than that specific value.

Conclusion for Card 70

• Do not reject null hypothesis due to calculated test statistic and p-value considerations.
• Sample mean salary from 61 employees is $56,500 with a standard deviation of $3,750.
• Test statistic calculated as 3.12 with 60 degrees of freedom signifies a right-tailed test.
• The p-value associated with t = 3.12 is less than 0.005, indicating strong evidence against null hypothesis if true.

Conclusion for Card 74

• Rebecca tests whether the mean market value of houses is greater than $250,000.
• Sample of 35 houses gives a mean market value of $259,860 with a standard deviation of $24,922.
• t-statistic calculated as approximately 2.34 with 34 degrees of freedom.
• At a significance level (α) of 0.05, Rebecca should assess whether to reject or fail to reject the null hypothesis based on p-value interpretation, which should be obtained using the t-distribution table.