Statistics and Probability: Testing a Claim

After this lesson, you should be able to do the following:
- State appropriate hypotheses for a significance test about a population parameter.
- Interpret a P-value in context.
- Make an appropriate conclusion for a significance test based on a P-value.

Confidence intervals are one of the two most common types of statistical inference.
- Usage of Confidence Intervals:
- Used when the goal is to estimate a population parameter.
- Significance Test:
- The second common type of inference focuses on testing a claim about a parameter.
Definition of a Significance Test:
- A formal procedure for using observed data to decide between two competing claims (hypotheses).
- Claims are statements about a parameter, such as the population proportion $p$ or the population mean $\boldsymbol{ ext{µ}}$ .

Important Concepts:
- Begins with a careful statement of claims to compare.
- Null Hypothesis ( $H_0$ ):
- The claim we seek evidence against in statistical testing.
- Alternative Hypothesis ( $H_a$ ):
- The claim we hope or suspect to be true instead of the null hypothesis.

Null Hypothesis Form:
- $H_0: ext{parameter} = ext{null value}$
Alternative Hypothesis Types:
- One-sided Alternative Hypothesis:
- Either H_a: ext{parameter} < ext{null value} or H_a: ext{parameter} > ext{null value}.
- Two-sided Alternative Hypothesis:
- $H_a: ext{parameter} <br>eq ext{null value}$ .
Definition of One-sided Alternative Hypothesis:
- States that a parameter is either greater than or less than the null value.
Definition of Two-sided Alternative Hypothesis:
- States that a parameter is different from the null value (can be either greater than or less than).

Key Considerations:
- Hypotheses should express the hope or suspicion before viewing data.
- Hypotheses refer to a population, not a sample.

P-value Definition:
- The P-value of a test is the probability of obtaining evidence for the alternative hypothesis $H_a$ as strong as or stronger than what was observed, under the assumption that the null hypothesis $H_0$ is true.

Final Step:
- Draw a conclusion about the competing claims based on the strength of evidence against the null hypothesis, measured by the P-value.
- Interpreting P-values:
- Small P-values:
  - Provide convincing evidence for $H_a$ (imply results unlikely if $H_0$ were true).
- Large P-values:
  - Fail to provide convincing evidence for $H_a$ (imply results likely to occur by chance if $H_0$ is true).
Decision-making:
- If P-value is small: Reject $H_0$ ; conclude there is convincing evidence for $H_a$ in context.
- If P-value is large: Fail to reject $H_0$ ; conclude there is not convincing evidence for $H_a$ in context.

Context: Study from the San Gabriel Valley Tribune claims that a majority of couples prefer to tilt their heads to the right when kissing.
Data Collected:
- A random sample of 124 kissing couples found 83 couples (approx. $p = rac{83}{124} = 0.669$ ) tilted to the right.
Questions to Address:
1. State appropriate hypotheses for the significance test.
- Parameter of Interest: Proportion of kissing couples who tilt their heads to the right.
- $H_0: p = 0.5$ (null hypothesis — 50% tilt to the right).
- H_a: p > 0.5 (alternative hypothesis — more than 50% tilt to the right).
1. P-value for the test is 0.0001. Interpret this in context.
- This P-value indicates that the probability of observing at least 83 couples tilting to the right out of 124 is extremely low (if the null hypothesis were true).
1. Conclusion:
- After evaluating the small P-value, reject $H_0$ ; conclude there is strong evidence that more than 50% of couples prefer to tilt to the right (in context).