Statistical Tests - T-Test and Z-Test Notes

Chapter 9 Notes: Statistical Tests Overview

Single Sample Z-Test

  • Formula: z = \frac{M - \mu}{\sigma_M}
    • Where:
    • M = sample mean
    • \mu = population mean
    • \sigma_M is the standard deviation of the sample mean

Single Sample T-Test

  • Purpose: Used when the population standard deviation (σ) is unknown.
  • Estimated Standard Error:
    • Formula:
      s_M = \frac{s}{\sqrt{n}}
    • Where:
      • s = sample standard deviation
      • n = sample size
  • Formula for T-Test:
    t = \frac{M - \mu}{s_M}

Steps of the T-Test

  1. Set the hypotheses (null H0 and alternative H1 ).
  2. Set the decision criteria (select alpha level, directionality, critical t values).
  3. Compute the test statistic ( t ).
  4. Locate the test statistic in the sampling distribution and compare to critical value(s).
  5. Make a statistical decision (reject or fail to reject H_0 ).
  6. State the conclusion and describe the nature of the effect, if present.

Degrees of Freedom

  • For a single sample t-test:
    df = n - 1

Finding Critical T-Values

  • Use Table B2 for the t-distribution.
  • Example:
    • Sample size: n = 4
    • One-tailed test; \alpha = .05
    • df = 3 ; critical value t = 2.353
  • Two-tailed test example:
    • Sample size: n = 9 ; df = 8 ;
    • Critical values t = \pm 2.306

Influence of Sample Variance and Size on T-Test Outcome

  • Larger sample variance → Larger standard error → Smaller t → Lower power
  • Smaller sample variance → Smaller standard error → Larger t → Higher power
  • Smaller sample size → Larger standard error → Smaller t → Lower power
  • Larger sample size → Smaller standard error → Larger t → Higher power
  • Power: Probability of correctly rejecting the null hypothesis.

Assumptions of the Single Sample T-Test

  1. Independent observations.
  2. The population sampled must be normally distributed.

When to Use Z-Test vs. T-Test

  • Check for variability information for the population:
    • If information is present (e.g., SS , \sigma^2 , or \sigma ), use a z-test.
    • If not, use a t-test.

Chapter 9 Examples

Example 1: Police Officer Work Hours
  • Claim: Officers work significantly more than 40 hours.
  • Data: 9 officers, hours worked: 43, 45, 48, 41, 45, 50, 50, 53, 48
  • Calculations:
    • n=9 , s^2=14.5 , M=47 , s=3.81
  • Hypotheses:
    • H_0: \mu = 40
    • H_1: \mu > 40
    • \alpha = .05 , one-tailed
    • df = 8
    • Critical t = +1.860
    • Test statistic:
      t = \frac{47 - 40}{1.27} = 5.51
  • Result: Reject H_0 ; officers work significantly more than 40 hours, t(8) = 5.51, p < .05 .
Example 2: Study Skills Training Program
  • Claim: Students in the program perform differently than the rest of the class.
  • Data: 25 freshmen, past class mean = 74, program mean = 78, SS=2400 .
  • Calculations:
    • n=25 , s^2=100 , M=78 , s=10
  • Hypotheses:
    • H_0: \mu = 74
    • H_1: \mu > 74
    • \alpha = .05 , one-tailed
    • df = 24
    • Critical t = +1.711
    • Test statistic:
      t = \frac{78 - 74}{2} = 2
  • Result: Reject H_0 ; students in the program performed better, t(24) = 2.0, p < .05 .
Example 3: Humidity Effect on Rats' Eating
  • Claim: Humidity affects eating behavior.
  • Data: $ n=100 $, sample mean = 18.7, population mean = 21, SS=2475 .
  • Calculations:
    • s^2=25 , s=5
  • Hypotheses:
    • H_0: \mu = 21
    • H_1: \mu \neq 21
    • \alpha = .05 , two-tailed
    • df = 99 , critical t = \pm 2.00
    • Test statistic:
      t = \frac{18.7 - 21}{0.5} = -4.6
  • Result: Reject H_0 ; humidity significantly decreases food consumption, t(99) = -4.6, p < .05 .