Hypothesis Testing Requirements

Overview

• Focus on requirements for hypothesis testing.
• Specifically examines requirements for means and proportions.

Requirements for Means

Z-Test for a Mean

• Key Requirements:
  • Testing a population mean.
  • Population is normally distributed or sample size ( n \geq 30 ).
  • Sample comes from a simple random sample.
  • Population standard deviation (( \sigma )) is known.
• Z-Score Formula:
  • ( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} )
  • ( \bar{x} ) is the sample mean, ( \mu ) is the population mean, ( \sigma ) is the standard deviation.
• Test Statistic: Used in the hypothesis test.
• Difference from T-Test: Z-Test uses known ( \sigma ).

T-Test for a Mean

• Key Requirements:
  • Testing a population mean.
  • Population is normally distributed or sample size ( n \geq 30 ).
  • Sample comes from a simple random sample.
  • Population standard deviation (( \sigma )) is unknown but sample standard deviation (( s )) is known.
• T-Score Formula:
  • ( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} )
  • ( \bar{x} ) is the sample mean, ( \mu ) is the population mean, ( s ) is the sample standard deviation.
• Test Statistic: Used in the hypothesis test.
• Difference from Z-Test: T-Test uses sample standard deviation, not population.

Requirements for Proportion Test

• Key Requirements:
  • Testing a population proportion.
  • Population is normally distributed or sample satisfies ( n \times p \times (1 - p) \geq 10 ).
  • Sample from a simple random sample.
• Z-Score for Proportion:
  • ( z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} )
  • ( \hat{p} ) is the sample proportion, ( p ) is the population proportion.
• Test Statistic: Used in the hypothesis test for proportions.

Choosing the Right Test

• Parameter Testing:
  • Determine if testing a mean or a proportion.
  • For Means:
    • Know population standard deviation? Use Z-Test.
      • Test Statistic: ( \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} )
    • Unknown population standard deviation but know sample standard deviation? Use T-Test.
      • Test Statistic: ( \frac{\bar{x} - \mu}{s / \sqrt{n}} )
  • For Proportions:
    • Directly use proportion test.
      • Test Statistic: ( z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} )

Summary

• Distinction between Z and T tests is crucial due to different table values and graph shapes.
• Later videos will explore using calculators for these tests.