7.3 Confidence Intervals for Proportions

Confidence intervals for proportions help estimate population values based on sample data. They provide a range of likely values for the true proportion, considering sample size and desired confidence level. This powerful tool allows businesses to make informed decisions.

To use confidence intervals effectively, certain conditions must be met. These include random sampling, large enough sample sizes, and proper interpretation of results. Understanding these concepts enables more accurate analysis and better decision-making in various business scenarios.

Confidence Intervals for Proportions

Confidence intervals for population proportions

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• Calculate confidence intervals for population proportions using the normal distribution approximation
• Formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
  • $\hat{p}$ represents the sample proportion (percentage of successes in the sample)
  • $z_{\alpha/2}$ is the critical value from the standard normal distribution based on the desired confidence level
    • $\alpha$ is the significance level, and $\alpha/2$ is used for two-sided confidence intervals (90%, 95%, 99%)
  • $n$ is the sample size (number of observations in the sample)
• Provides a range of plausible values for the true population proportion based on sample data

Conditions for normal distribution approximation

• Verify conditions for using the normal distribution approximation to construct confidence intervals for proportions
  1. Random and independent sample selection ensures unbiased representation of the population
  2. Large enough sample size satisfies the following criteria:
     • $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$
     • Ensures the sampling distribution of the sample proportion is approximately normal (central limit theorem)
• Meeting these conditions allows for accurate approximation and reliable confidence intervals

Margin of error for proportion intervals

• Calculate the margin of error to determine the precision of the confidence interval estimate
• Formula: Margin of error = $z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
  • Represents the range of values above and below the sample proportion where the true population proportion likely falls
  • Smaller margin of error indicates higher precision and narrower confidence interval
  • Larger sample sizes decrease the margin of error, holding other factors constant (confidence level, sample proportion)
• Quantifies the uncertainty associated with the confidence interval estimate

Interpretation of proportion confidence intervals

• Interpret the meaning and implications of a confidence interval for a population proportion in a business context
• Example: 95% confidence interval for the proportion of defective products (0.03, 0.07)
  • 95% confidence that the true proportion of defective products in the population falls between 3% and 7%
  • Repeated sampling would result in 95% of the confidence intervals containing the true population proportion
• Narrower intervals indicate more precise estimates, while wider intervals suggest less precision
• Higher confidence levels (99%) yield wider intervals, while lower confidence levels (90%) produce narrower intervals
• Confidence level represents the long-run probability of the interval containing the true population proportion
• Use confidence intervals to make data-driven business decisions and assess the reliability of sample estimates