2 Sample interval of proportions

Statistical Testing Assumptions

  • Assumption of Simple Random Sampling (SRS)

    • The first assumption is that samples taken are simple random samples.

    • Ensures unbiased representation from the population.

  • Population Size Requirements

    • Population size for both groups must be larger than 10 for meaningful analysis.

    • Denote populations as greater than ten for both group one and group two.

Changes in Statistical Testing Techniques

  • From Significant Tests to Non-Pooled Proportions

    • In significant tests, the null hypothesis states the samples come from the same population.

    • Now, hypotheses are tested independently since there is no presumption of common population.

    • Avoid pooling data as the underlying population assumption is no longer valid.

Conditions for Validity

  • Calculating Sample Sizes

    • Ensure sufficient sample sizes for both p-hat calculations:

      • n1 * p-hat1 ≥ 10

      • n1 * q-hat1 ≥ 10

      • n2 * p-hat2 ≥ 10

      • n2 * q-hat2 ≥ 10

  • Significant vs Confidence Interval

    • In significance tests, data may be pooled; however, in confidence intervals, data is kept separate.

Example Analysis: Parental Influence on Teen Smoking

  • Study Overview

    • A study published by the American Academy of Pediatrics examined the relationship between parental attitudes and teen smoking.

    • Initial group consisting of students who have never smoked and their parental attitudes was surveyed.

  • Findings after Two Years

    • Among students whose parents disapproved of smoking, 54 out of 284 became smokers.

    • Among those whose parents were lenient, 11 out of 41 became smokers.

  • Analysis Approach

    • Use a 95% confidence interval to analyze whether a difference exists between the two parental attitude groups.

  • Confidence Interval Estimation

    • Calculate proportions and compare using:

      • P-hat calculations from both samples and evaluating differences.

      • Calculate margin of error to assess range of potential differences.

  • Final Conclusion

    • If zero is included within the confidence interval, it indicates no significant difference in smoking rates based on parental attitudes.

Personal Anecdotes and Vaping Trend Discussion

  • Impact of Vaping on Perception of Smoking

    • Discussed differences between smoking and vaping among youth.

    • Highlighted the cultural shift and changes in behavior and norms surrounding smoking.

Two Sample Proportions Analysis in Voter Support

  • Case Study: Voter Support Before and After Scandal

    • Examined a situation where voter support dropped from 54% to 51% after a candidate scandal.

  • Statistical Analysis Approach

    • Confirmed whether this is a two-sample problem focusing on the proportions.

    • Analyzed sample sizes for validity of statistical assumptions.

  • Testing Procedure

    • Calculated the z-score for differences in proportions:

      • Evaluated how significant the change is using p-values and alpha levels.

  • Conclusion Interpretation

    • If the p-value is less than the alpha level, we would reject the null hypothesis,

    • Leading to a conclusion that voter support indeed decreased after the candidate's scandal.