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4.6.1 Sampling Distribution of Proportion

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    4.6 Sampling Distribution of Proportion
    
    4.6.1 Sampling Distribution of Proportion 4.6.1 Sampling Distribution of Proportion
    Table of contents
    
    Introduction
    
    Understanding Sample Proportions
    
    Properties of Sampling Distribution
    
    Practical Applications
    
    Example: Medical Survey
    
    Conclusion
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    4.7.1 Distribution of Sample Variance
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    5.2.4 Determining the Sample Size
  - 5.3 Confidence Interval for Proportion
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    5.3.1 Confidence Interval for Proportion
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    5.8.1 Hypothesis Testing for Population Proportion
    
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    7.5.1 Introduction to Retailing
    
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    7.5.4 Retail Formats
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    7.6.1 Introduction to D'Light
    
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    8.2.1 Macro Communication
    
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    8.4.2 Sales Process in B2B Market
    
    8.4.3 Sales Process in B2C Market
  - 8.5 Sales Force Management
    8.5 Sales Force Management
    
    8.5.1 Designing Effective Sales Force
    
    8.5.2 Managing the Sales Force

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.

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