Lecture Notes: Testing Hypotheses About a Single Mean

Key Concepts

• Objective: Make an inference about a population mean ( \mu ) given a sample.
• Comparison: Population mean ( \mu ) vs. hypothesized mean ( \mu_0 ).
• Assumptions: Continuous variable ( x ) is normally distributed with mean ( \mu ) and variance ( \sigma^2 ).

Test Types

1. When Population Variance is Known

• Test Used: Z-test

2. When Population Variance is Unknown

• Test Used: T-test
• Common Scenario: More frequent as population variance is rarely known.

Hypotheses Forms

• Two-sided:
  • Null Hypothesis (( H_0 )): ( \mu = \mu_0 )
  • Alternative Hypothesis (( H_a )): ( \mu \neq \mu_0 )
• One-sided:
  • If testing ( \mu < \mu_0 ), ( H_0: \mu \geq \mu_0 )
  • If testing ( \mu > \mu_0 ), ( H_0: \mu \leq \mu_0 )
  • Note: Null hypothesis is opposite in direction from the alternative.

Test Statistic: Z-Test

• Formula: [ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} ]
  • ( \bar{x} ): Sample mean
  • ( \mu_0 ): Theorized value
  • ( \sigma ): Standard deviation (known)
  • ( n ): Sample size
• Interpretation: Z-statistic follows the standard normal distribution under null hypothesis.

Example: IQ of LA Residents

• Objective: Check if mean IQ differs from national average of 100.
• Known Variance: 225 (standard deviation = 15)
• Sample: 7 LA residents
• Result:
  • Sample mean ( = 99.5 )
  • Z-statistic ( = -0.07559 )
  • P-value ( = 0.93974 )
  • Conclusion: Do not reject the null hypothesis.

Conclusion Statements

• Components:
  • Restate the test's purpose.
  • Report sample statistics.
  • Indicate statistical significance.
  • Specify directionality.
  • Report the comparator and p-value.

Example: Coin Flips by PM510 Students

• Objective: Compare students' streaks of heads/tails to theoretical expectations.
• Population Mean and SD: 6.977432 and 1.7926
• Sample Mean: 5.112903
• Result:
  • Z-statistic ( = -8.189956 )
  • P-value < 0.00001
  • Conclusion: Reject null hypothesis.

T-statistic

• Used when: Population variance unknown.
• Formula: Similar to Z, but uses sample variance.
• Distribution: Follows Student's t-distribution with ( n-1 ) degrees of freedom.

Example: CD4 Levels in HIV Patients

• Objective: Check if mean differs from 400 (cutoff)
• Sample: 10 patients
• Result:
  • Sample mean ( = 305.5 )
  • T-statistic ( = -2.98835 )
  • P-value ( = 0.01524 )
  • Conclusion: Mean is statistically different from 400.

Example: White Blood Cell Count in Leukemia

• Objective: Check if mean > 7500 (cutoff)
• Challenge: Data not normally distributed.
• Solution: Log transformation
• Result:
  • Sample mean (log transformed) ( = 0.6765 )
  • T-statistic ( = -1.753 )
  • Adjust for direction with one-sided test: P-value ( = 0.9565 )
  • Conclusion: Do not reject null hypothesis.

Important Note on One-sided Tests

• SPSS Caution: SPSS output for one-sided tests can be misleading.
• Approach: Use two-sided p-value as a starting point and adjust based on alignment of observed vs. hypothesized means.

Summary

• Z-test: For known variance, compare ( \mu ) and ( \mu_0 ).
• T-test: For unknown variance, uses sample data to estimate.
• P-value Significance: Helps conclude whether to accept or reject the null hypothesis.