To determine a confidence interval for a population proportion using the sample proportion (\( \hat{p} \)), we typically check the assumptions you mentioned to ensure the validity of the approximation.
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Random Sampling: The sample must be drawn randomly from the population to allow the results to be generalized to that population.
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Independence: The values in the sample must be independent of each other which is generally ensured by random sampling and selecting a sample size less than 10% of the population (for finite populations).
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Sufficient Sample Size: The conditions \( n\hat{p} \geq 10 \) and \( n(1 - \hat{p}) \geq 10 \) are necessary to use the normal approximation to the binomial distribution. Here, \( n \) is the sample size and \( \hat{p} \) is the sample proportion of successes. These ensure that there are enough successes and failures in the sample to reliably estimate the population proportion.
Once these assumptions are satisfied, the confidence interval for the population proportion can be calculated using the formula:
\[ \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Where:
- \( \hat{p} \) = sample proportion
- \( Z \) = z-score corresponding to the desired confidence level (for example, for a 95% confidence level, \( Z \) is approximately 1.96)
- \( n \) = sample size
In summary, it's crucial to verify these assumptions before applying the method for calculating confidence intervals to ensure the results are valid.
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