Chapter 7 Notes: Hypothesis Testing I

Hypothesis Testing: One-Tailed Tests – Example 2

• The education department at a university is under investigation for potential grade inflation.
• The new chair wants to determine if education majors have higher GPAs compared to students in general.

Establishing the Critical Region: One-Tailed vs. Two-Tailed Tests ($\alpha = 0.05$)

• Two-Tailed Test:
  • Critical values: $Z_{critical} = \pm 1.96$
  • 95% of the total area lies within these critical values.
• One-Tailed Test (Upper Tail):
  • Critical value: $Z_{critical} = +1.65$
  • 95% of the total area is to the left of this critical value.
• One-Tailed Test (Lower Tail):
  • Critical value: $Z_{critical} = -1.65$
  • 95% of the total area is to the right of this critical value.

One-Tailed Tests: Steps

• Step 1: State assumptions and meet test requirements.
• Step 2: Formulate Null and Alternative Hypotheses:
  • Null Hypothesis ($H_0$): $\mu = 2.7$
  • Alternative Hypothesis ($H_1$): \mu > 2.7
• Step 3: Establish the Critical Region
  • Set the significance level ($\alpha$): e.g., $\alpha = 0.05$
  • Determine the critical value: $Z_{critical} = +1.65$ (for a one-tailed test), which is different from $\pm 1.96$ (for a two-tailed test).

One-Tailed Tests: Decision and Interpretation

• Step 4: Calculate the Test Statistic:
  • $Z_{obtained} = 4.62$
• Step 5: Make a Decision and Interpret Results
  • Compare $Z<em>{critical} = 1.65$ with $Z</em>{obtained} = 4.62$
  • Conclusion: The GPA of education majors is significantly higher than that of the general student body.

One-Tailed vs. Two-Tailed Tests: Critical Values

• Different critical Z scores are used for one-tailed and two-tailed tests.
• Reference table for finding critical Z scores for one-tailed tests (single sample means):
  • Alpha = 0.10: Two-Tailed = $\pm 1.65$, Upper Tail = $+1.29$, Lower Tail = $-1.29$
  • Alpha = 0.05: Two-Tailed = $\pm 1.96$, Upper Tail = $+1.65$, Lower Tail = $-1.65$
  • Alpha = 0.01: Two-Tailed = $\pm 2.58$, Upper Tail = $+2.33$, Lower Tail = $-2.33$
  • Alpha = 0.001: Two-Tailed = $\pm 3.29$, Upper Tail = $+3.10$, Lower Tail = $-3.10$

Type I and Type II Errors

• Analogy to a Legal Trial:
  • Defendant Innocent ($H_0$ is True):
    • Convict: Type I error
    • Acquit: OK
  • Defendant Guilty ($H_1$ is True):
    • Convict: OK
    • Acquit: Type II error

Type I and Type II Errors: Decision Making and the Null Hypothesis

• Decision:
  • Reject $H<em>0$ when $H</em>0$ is True: Type I error ($\alpha$ error)
  • Fail to Reject $H<em>0$ when $H</em>0$ is True: OK
  • Reject $H<em>0$ when $H</em>0$ is False: OK
  • Fail to Reject $H<em>0$ when $H</em>0$ is False: Type II error ($\beta$ error)

Example 3: t-test for a Small Sample

• A new chair wants to know if education majors have a different average GPA than students in general.
• The average GPA of all students is known: 2.7.
• A random sample of 20 education majors is taken, with:
  • Sample average GPA: 3.0
  • Sample standard deviation: 0.7

Example 3: Small Sample Details

• Population parameter (average GPA for all students): 2.7
• Sample statistics for education majors:
  • $\mu_0 = 2.7$
  • $\bar{x} = 3.0$
  • $s = 0.7$
  • $N = 20$

t-Distribution vs. z-Distribution (Visualization)

• Illustration comparing the shapes of the t-distribution and the z-distribution.

Hypothesis Testing: t-Test Steps

• Step 1: Assumptions and Requirements
  • Use Student’s t-distribution.
• Step 2: Hypotheses
  • $H_0: \mu = 2.7$
  • $H_1: \mu \neq 2.7$
• Step 3: Establish the Critical Region
  • Small sample size (N = 20).
  • Use the t-distribution table.
  • Degrees of freedom: $df = N - 1 = 19$
  • If $\alpha = 0.05$, then $t_{critical} = \pm 2.093$ (two-tailed test).

t-Test: Decision

• Step 4: Compute the Test Statistic (t).
• Step 5: Make a Decision
  • $t_{critical} = \pm 2.093$
  • $t_{obtained} = 1.91$
  • Conclusion: Cannot reject $H_0$ at $\alpha = 0.05$. The average GPA of education majors is not significantly different from the general student body.

Hypothesis Testing for a Single Sample Proportion – Example 4

• A new chair wants to know if education majors have a different proportion of receiving straight ‘A’ in core courses than students in general.
• Known: 20% of all students receive straight “A” in core courses.
• Random sample: 117 education majors, 35 received straight “A” in core courses.

Hypothesis Testing for a Single Sample Proportion – Details

• Population parameter: 20% of all students receive straight “A” (P0 = 0.2).
• Sample from education majors:
  • N = 117
  • 35 students with straight “A”s
  • Sample proportion: $P_s = \frac{35}{117} \approx 0.3$

Hypothesis Testing for A Single Sample Proportion - Step 1

• Step 1: Meet Test Requirements
  • A random sample.
  • Variable is nominal-level (proportion of students receiving straight “A”).
  • Sampling distribution is approximately normal for large samples (N > 30).

Hypothesis Testing for A Single Sample Proportion - Steps 2 & 3

• Step 2: Null & Alternative Hypotheses
  • $H<em>0: P</em>u = 0.2$ (i.e., $P<em>u = P</em>0 = 0.2$)
  • $H<em>1: P</em>u \neq 0.2$ (i.e., $P<em>u \neq P</em>0$)
• Step 3: Establish the Critical Region
  • Set $\alpha$: e.g., $\alpha = 0.05$
  • $Z_{critical} = \pm 1.96$ (two-tailed test)

Hypothesis Testing for A Single Sample Proportion - Steps 4 & 5

• Step 4: Compute the test statistic.
• Step 5: Make a Decision
  • $Z_{critical} = \pm 1.96$
  • $Z_{obtained} = 2.704$
  • Conclusion: Reject $H_0$. Education majors have a significantly different proportion of receiving straight ‘A’ in core courses than the general student body.

Lab and Homework

• Lab 5 exercise (ungraded): 7.5 (p. 204)
• HW6 (graded): 7.2 (a), 7.9, 7.12 (pp. 204-205)