Quiz Review 8.1 and 8.2 Key

Page 1: Confidence Interval Calculation for Proportion

  • Background: We aim to estimate the true proportion of all adult heterosexuals who have had multiple partners, based on data from the National AIDS Behavioral Surveys where 170 out of 2673 sampled individuals reported having multiple partners. This gives a sample proportion (P) of 0.0636.

Conditions for Inference Procedure

  1. Simple Random Sample (SRS): We assume this sample is an SRS representative of all adult heterosexuals.

  2. Sample Size Requirements: Assessing the sample size, we calculate:

    • Total adult heterosexuals = 2673 > 10(0.0636) = 0.636 (met condition)

    • For successes: np = 2673 * 0.0636 = 170.11

    • For failures: n(1-p) = 2673 * (1-0.0636) = 2502.89 Thus, both successes (170) and failures (2502) exceed 10, fulfilling the requirement.

Calculation of Standard Error

  • The standard error (SEp) is computed as: [ SE_p = \sqrt{\frac{P(1-P)}{n}} = \sqrt{\frac{0.0636(0.9364)}{2673}} ]

Confidence Interval Formula

  • The confidence interval is calculated using the formula: [ CI = P \pm Z^* (SE) ] where:

    • Z* for 99% confidence = 2.576

    • Calculating: [ CI = 0.0636 \pm 2.576 * 0.0122 = (0.0514, 0.0758) ]

Conclusion

  • Interpretation: With 99% confidence, we conclude that the proportion of all adult heterosexuals who have had multiple partners in the last year is between 5.14% and 7.58%.

Page 2: Understanding Confidence Intervals

  • General Observation: A confidence interval (CI) does not guarantee that each individual interval covers the true parameter; rather, 95% of CIs from repeated sampling would capture the true parameter.

Confidence Interval Interpretations

The Good:

  1. 95% CI for the mean SAT score of HS Seniors is between 452 and 470.

  2. After multiple SRS of size 30, 95% of the intervals will capture the true mean.

  3. Confidently stating that the population parameter lies within the interval.

  4. The CI method will capture the true population parameter 95% of the time.

The Bad and Ugly:

  1. Incorrectly suggesting that 95% of California HS seniors fall within the score range.

  2. There's no direct probability associated with a specific interval containing the parameter.

  3. Misrepresenting that the population parameter is deterministically in the interval.

  4. Incorrectly stating a fixed chance for the mean falling within that range.

Key Takeaway

  • The population parameter is unknown; we estimate its value and hope our sample accurately reflects the entire population.

Page 3: Correct Interpretation of Confidence Intervals

  • A 95% CI for daily sleep time of dogs indicates:

    • Directly stating a CI as capturing a percentage of dogs or sample means is incorrect as it misrepresents the meaning of confidence intervals.

Valid Statements about CI Interpretations:

  1. If the procedure is repeated, 95% of intervals would contain the population mean.

  2. Confidently stating the average lies within the calculated bounds.

  3. 95% of sample means would lie within the specified range if repeatedly sampled.

  4. The true average sleep time is accurately reflected by the CI.

Page 4: Evaluating Confidence Interval Statements

  • The following statements should be evaluated as correct or incorrect:

  1. Incorrect: 95% of the time applies to confidence intervals, not raw data measurements.

  2. Incorrect: Not all constructed intervals will contain the true mean, but we expect 95% to.

  3. Correct: This accurately expresses confidence about the population mean.

  4. Incorrect: Fixed intervals signify our method, not the specific interval’s content.

  5. Incorrect: There's no definitive probability for mean numbers falling within ranges.

  6. Correct: Correctly rationalizes the outcome of repeated random sampling.

  7. Incorrect: No absolute confidence in the exact value of the mean.

  8. Correct: A proper CI implies plausible values for estimating the population parameter.

  9. Correct: Correctly conceptualizes probability in terms of the CI associated level.

  10. Correct: Properly contextualized use of probability in repeating sample surveys.

Page 5: Estimating Proportion with CI

  • Estimation Example: Calculate the proportion of EKHS students having seen 'The Office'. We establish a sample size of 100 and determine:

    • Sample DEFINITION: 10% assume seeing the show.

    • Calculate proportion including margin of error yielding a CI for P as:

    • Resulting interval: (0.535, 0.725), hence 95% confident this captures the true proportion.

Page 6: Determining Sample Size for CI

  • Choosing Sample Size: For estimating a parameter within a certain margin of error, follow:

    • Scenario: Need 95% confidence with a margin of error ≤ 0.03 under assumptions of SRS.

    • Using the formula: [ n = \frac{(Z^\ast)^2 \cdot p(1-p)}{ME^2} ] to find the required n.

Practical Example

  • For candidates Chavez and Flynn:

    • Desired margin of error = 0.03

    • Back calculating gives us the needed sample size of approximately 1068 to maintain 95% confidence

Page 7: Practical Sample Size Scenarios

  1. Mayor Candidate Comparison: Assessing required sample sizes:

    • Gustafson's Voting Proportion: Using conservative p to estimate n at 1068 for 95% confidence and 0.03 margin.

  2. Nightclub Proposal: For a nightclub targeting under-21 patrons with a suspected proportion of p=0.75, calculating the needed sample size for a 90% CI with a low margin of error yields approximately 318 surveyed individuals.