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{critical} = 1.65 with Z{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 H0 when H0 is True: Type I error ($\alpha$ error)
  • Fail to Reject H0 when H0 is True: OK
  • Reject H0 when H0 is False: OK
  • Fail to Reject H0 when H0 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
  • H0: Pu = 0.2 (i.e., Pu = P0 = 0.2)
  • H1: Pu \neq 0.2 (i.e., Pu \neq P0)
• 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)