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 $H<em>0$ and alternative $H</em>1$).
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 .