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4.6.1 Sampling Distribution of Proportion

Introduction

  • Purpose: Understanding the sampling distribution of the sample proportion (p̄) and its implications for statistical inference.

Understanding Sample Proportions

  • Definition: The sample proportion p̄ is a point estimator for the population proportion p. It is calculated as p̄ = x/n, where x is the number of elements in the sample with a characteristic of interest.
  • Example: In a customer satisfaction survey, if 17 out of 36 customers are satisfied, the sample proportion p̄ is 0.47.

Properties of Sampling Distribution

  • Expected Value: The expected value of p̄ is the population proportion p, making p̄ an unbiased estimator of p.
  • Standard Deviation (Standard Error): The standard deviation of the sampling distribution of p̄ is calculated as √(p(1-p)/n).
  • Shape of the Distribution: The distribution of p̄ approaches normality as the sample size n increases, a result supported by the Central Limit Theorem when n is large enough (usually n > 30).

Practical Applications

  • Probability Estimations:
  • Example: Estimating the likelihood that the sample proportion will be within a certain range of the population proportion.
  • Calculation involves determining the standard error and using the normal approximation to estimate probabilities.

Example: Medical Survey

  • Scenario: A survey aims to estimate the proportion of patients receiving unnecessary medical treatments.
  • Survey Details: A sample size of 150 is considered large enough to use the normal approximation.
  • Probability Questions:
  • The probability that the sample proportion is within a specific interval around the population proportion.
  • The probability that the sample proportion exceeds a certain threshold.

Conclusion

  • Importance: The sampling distribution of proportion is crucial in planning and interpreting surveys and experiments where proportion estimates are required.
  • Utility: Helps in determining the adequacy of sample sizes and in making probabilistic predictions about survey outcomes based on sample data.