Chapter 7 Notes: Hypothesis Testing I

Chapter 7: Hypothesis Testing I: The One-Sample Case (Part 1)

Overview of Hypothesis Testing

• Hypothesis testing is designed to detect significant differences, i.e., differences that do not occur by random chance.
• It involves comparing a population with an unknown parameter against a population with a known parameter.
• Sometimes, a specific large group ("subpopulation") is compared against a whole population.
• This chapter focuses on the one-sample case, where a random sample is drawn from the population with the unknown parameter.

Examples of Hypothesis Testing

• Do residents in Riverside have a different average household income from all residents in California?
• Do college athletes at UCR have a different average GPA from the student body as a whole?
• Are patients using a new drug more likely to experience a heart attack?

An Example: Test Different GPA

• A new chair of the education department at a university wants to know if education majors have a different average GPA than students in general.
• The average GPA of all students is known to be 2.7.
• A random sample of 117 students is drawn from all education majors. This sample has an average GPA of 3 and a standard deviation of 0.7.
• The average GPA of all education majors is unknown.

Example Data

• Population mean GPA (all students): m_0 = 2.7
• Sample mean GPA (education majors): \bar{x} = 3.0
• Sample standard deviation (education majors): s = 0.7
• Sample size: N = 117

Significance of the Difference

• There is a difference between the sample statistic (3.0) for 117 education majors and the population parameter (2.7) for all students.
• It appears that education majors may have higher GPAs.
• However, there is uncertainty because we are working with a random sample.

Two Possibilities

• The observed difference may have been caused by random chance.
  • All education majors have the same population mean GPA as all students (2.7).
  • The sample mean (3.0) from education majors occurred by chance.
• The difference is real (or "significant").
  • Education majors have a different population mean GPA than all students (2.7).

The Five-Step Model for Hypothesis Testing

1. Make assumptions and meet test requirements.
2. State null & research hypotheses (H0 and H1).
3. Select the sampling distribution and establish the critical region.
4. Compute the test statistic (e.g., z score for a large sample).
5. Interpret results and make a decision.

Step 1: Make Assumptions and Meet Test Requirements

• A random sample (the sample of 117 was randomly selected from all education majors).
• Variable being tested is Interval-Ratio (GPA is an interval-ratio variable so the mean is an appropriate statistic).
• Sampling distribution is a normal distribution (since this is a large sample, N > 30).

Step 2: State the Null and Research Hypotheses

• Null Hypothesis (H_0):
  • H_0 always states there is no significant difference.
• Research (or alternative) hypothesis (H_1):
  • The difference is real (i.e., not caused by random chance).
• One (and only one) of these explanations must be true. Which one?

Step 2: Null Hypothesis (H_0)

• H0: \mu = 2.7 (i.e., \mu = \mu0 = 2.7)
  • All education majors have the same population mean as all students.
  • The difference between the population mean (2.7) and the sample mean (3.0) is caused by random chance.

Step 2: Research Hypothesis (H_1)

• H1: \mu \neq 2.7 (i.e., \mu \neq \mu0)
  • All education majors have a different population mean GPA than all students (2.7).
  • The difference between the sample mean (3.0) of 117 education majors and the population mean (2.7) of all students reflects a real difference between the two populations.

Step 3: Select Sampling Distribution and Establish the Critical Region

• For a large sample (N > 30), use the Z score formula and the normal curve table.
• Setting Alpha (\alpha, the indicator of “rare” events) (e.g., \alpha = 0.05; \alpha = 0.01).
• Find the corresponding Z (critical) (e.g., \pm 1.96; \pm 2.58).
• Beyond the Z (critical) is the Critical Region.

Step 3: Critical Region and Decision Rule

• If the Z score (obtained) is beyond the Z (critical) and in the critical region, the chance of having that difference given H_0 is true will be very small (e.g., < \alpha = 0.05).
  • We reject H0 at the \alpha level and support H1: the observed difference is statistically significant.
• If the Z score (obtained) does not fall in the critical region, the chance of having that difference given H_0 is true will be relatively large (e.g., > \alpha = 0.05).
  • We fail to reject H_0 at the specific \alpha level: the observed difference is not statistically significant.

Step 4: Compute the Test Statistic

• Assuming H_0 being true, use an appropriate formula to calculate the Z score.
• Z_{obtained} = 4.62
• This is beyond Z_{critical} = \pm 1.96.

Step 5: Interpret Results and Make a Decision

• Compare Z{critical} = \pm 1.96; Z{obtained} = 4.62
• The obtained Z score falls in the Critical Region, so we reject the H_0.
  • If H_0 were true, a sample outcome of 3.0 would be very unlikely.
  • Therefore, H_0 must be false and should be rejected at the level of \alpha = 0.05.
• Conclusion: The average GPA of all education majors is significantly different from that of the general student body.

Assignments

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