Hypothesis testing notes

Hypothesis Testing

Hypotheses: Possible outcomes concerning a new design or claim.
- Example: A new engine design to reduce particulate matter.
  - Outcome 1: Design reduces particulate matter.
  - Outcome 2: Design does not reduce particulate matter.
Null Hypothesis (H0): States that a parameter is equal to a specific value (\mu0).
Alternate Hypothesis (H1): States that the parameter differs from the value specified in the null hypothesis (\mu0).

Left-Tailed: The parameter is less than the value specified in the null hypothesis.
- Example: H1: \mu < \mu0
Right-Tailed: The parameter is greater than the value specified in the null hypothesis.
- Example: H1: \mu > \mu0
Two-Tailed: The parameter is not equal to the value specified in the null hypothesis.
- Example: H1: \mu \neq \mu0
One-Tailed Hypotheses: Left-tailed and right-tailed hypotheses.

Example 1: Cereal boxes labeled as containing 20 ounces. An inspector thinks the mean weight may be less.
- Null Hypothesis (H_0): \mu = 20
- Alternate Hypothesis (H_1): \mu < 20 (Left-tailed test)
Example 2: Last year's mean monthly rent was $800. A real estate agent believes this year's mean rent is higher.
- Null Hypothesis (H_0): \mu = 800
- Alternate Hypothesis (H_1): \mu > 800 (Right-tailed test)
Example 3: Standardized test scores have a mean of 70. Modifications are made, and an educator believes the mean may have changed.
- Null Hypothesis (H_0): \mu = 70
- Alternate Hypothesis (H_1): \mu \neq 70 (Two-tailed test)

Analogy to a Criminal Trial:
- Null hypothesis is like the defendant (assumed true/innocent).
- Data/evidence is presented.
- If evidence strongly contradicts the null hypothesis, we reject it (similar to finding the defendant guilty).
Sample Mean and Variation:
- Example Exam Score:
  - H_0: \mu = 80
  - H_1: \mu \neq 80
  - If sample mean \bar{x} = 78, it's close to 80, so the null hypothesis is plausible.
  - If sample mean \bar{x} = 50, it's far from 80, so we might reject the null hypothesis.
Key Question: How big does the difference need to be before we reject H_0?
- Hypothesis testing methods consider sample size and distribution spread.

Possible Outcomes:
- Reject the null hypothesis.
- Fail to reject the null hypothesis (do not accept the null hypothesis).
Reject the Null Hypothesis: Conclude that the alternate hypothesis is true.
Fail to Reject the Null Hypothesis: There is not enough evidence to conclude the alternate hypothesis is true. The null hypothesis might be true.
- It means the evidence was not strong enough to reject it.
Example 1 (Cereal Boxes):
- H_0: \mu = 20
- H_1: \mu < 20
- If H_0 is rejected: Conclude the mean weight of cereal is less than 20 ounces.
Example 2 (Cereal Boxes):
- H_0: \mu = 20
- H_1: \mu < 20
- If H_0 is not rejected: There is not enough evidence to conclude the mean weight is less than 20 ounces.

Type I Error: Rejecting the null hypothesis when it is actually true.
Type II Error: Failing to reject the null hypothesis when it is actually false.
Correct Decisions:
- Rejecting the null hypothesis when it is false.
- Failing to reject the null hypothesis when it is true.
Example (Dean's Salary Study):
- H_0: \mu = 50,000
- H_1: \mu > 50,000
- Scenario 1: True mean is \mu = 50,000, and the Dean rejects H_0. This is a Type I error.
- Scenario 2: True mean is \mu = 55,000, and the Dean rejects H_0. This is a correct decision.
- Scenario 3: True mean is \mu = 55,000, and the Dean does not reject H_0. This is a Type II error.