Hypothesis testing notes

Hypothesis Testing

Null and Alternate Hypotheses

• 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 ($H<em>0$): States that a parameter is equal to a specific value ($\mu</em>0$).

• Alternate Hypothesis ($H<em>1$): States that the parameter differs from the value specified in the null hypothesis ($\mu</em>0$).

Types of Alternate Hypotheses

• Left-Tailed: The parameter is less than the value specified in the null hypothesis.

  • Example: $H<em>1: \mu < \mu</em>0$

• Right-Tailed: The parameter is greater than the value specified in the null hypothesis.

  • Example: $H<em>1: \mu > \mu</em>0$

• Two-Tailed: The parameter is not equal to the value specified in the null hypothesis.

  • Example: $H<em>1: \mu \neq \mu</em>0$

• One-Tailed Hypotheses: Left-tailed and right-tailed hypotheses.

Examples of Stating Null and Alternate 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)

Reasoning in Hypothesis Testing

• 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.

Stating Conclusions in Hypothesis Testing

• 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 and Type II Errors

• 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.

Flounder Example: Stating Null and Alternate Hypotheses

• Sample of 100 flounder, sample mean weight is 21.5 grams.
• Test how strong the evidence is that the mean weight differs from 20 grams.
• $H_0: \mu = 20$
• $H_1: \mu \neq 20$ (Two-tailed alternate hypothesis)