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.

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.

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).

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.

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

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)

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

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

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).

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.

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.

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

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).

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)

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.