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Confidence Intervals

A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true population parameter. It provides an estimate of the parameter along with a measure of uncertainty.

Why Use Confidence Intervals?

  1. Precision: Provides a range rather than a single point estimate, offering more information about the parameter.
  2. Uncertainty: Quantifies the level of confidence in the estimate.
  3. Interpretability: Helps in decision-making by considering the possible range of values for a parameter.

Key Components of a Confidence Interval

  1. Point Estimate: The central value (e.g., sample mean) used as the best estimate of the population parameter.
  2. Margin of Error: The maximum expected difference between the point estimate and the true parameter.
  3. Confidence Level: The probability that the interval contains the true parameter, typically expressed as a percentage (e.g., 95%).

Common Confidence Levels:

  • 90% Confidence Level: Less precise, smaller margin of error.
  • 95% Confidence Level: Most commonly used.
  • 99% Confidence Level: More precise, larger margin of error.

Formula for Confidence Interval

For a Population Mean ($\mu$):

$$ \text{CI} = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}} $$

Where:

  • $\bar{x}$: Sample mean
  • $Z$: Z-score corresponding to the confidence level
  • $\sigma$: Population standard deviation (or sample standard deviation $s$ if $\sigma$ is unknown)
  • $n$: Sample size

For a Population Proportion ($p$):

$$ \text{CI} = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} $$

Where:

  • $\hat{p}$: Sample proportion
  • $Z$: Z-score corresponding to the confidence level
  • $n$: Sample size

Z-Scores for Common Confidence Levels

Confidence Level Z-Score ($Z$)
90% 1.645
95% 1.960
99% 2.576

Example 1: Confidence Interval for a Mean

A sample of 50 students has a mean test score of 75 with a standard deviation of 10. Calculate the 95% confidence interval for the population mean.

Solution:

$$ \text{CI} = \bar{x} \pm Z \cdot \frac{s}{\sqrt{n}} $$

$$ \text{CI} = 75 \pm 1.96 \cdot \frac{10}{\sqrt{50}} $$

$$ \text{CI} = 75 \pm 1.96 \cdot 1.414 \approx 75 \pm 2.77 $$

$$ \text{CI} = [72.23, 77.77] $$

The 95% confidence interval for the population mean is $[72.23, 77.77]$.

Example 2: Confidence Interval for a Proportion

In a survey of 200 people, 60% said they prefer coffee over tea. Calculate the 95% confidence interval for the population proportion.

Solution:

$$ \text{CI} = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} $$

$$ \text{CI} = 0.60 \pm 1.96 \cdot \sqrt{\frac{0.60 \cdot 0.40}{200}} $$

$$ \text{CI} = 0.60 \pm 1.96 \cdot \sqrt{0.0012} \approx 0.60 \pm 1.96 \cdot 0.0346 $$

$$ \text{CI} = 0.60 \pm 0.0679 $$

$$ \text{CI} = [0.532, 0.668] $$

The 95% confidence interval for the population proportion is $[0.532, 0.668]$.

Interpretation of Confidence Intervals

A 95% confidence interval means that if we were to take 100 random samples and calculate the CI for each, approximately 95 of them would contain the true population parameter.

Important Notes:

  • The CI does not imply that the true parameter has a 95% probability of lying within the interval. The interval either contains the parameter or it doesn’t.
  • Larger sample sizes result in narrower confidence intervals (more precision).
  • Higher confidence levels result in wider confidence intervals (more certainty).

Applications of Confidence Intervals

  1. Business: Estimating average customer satisfaction scores.
  2. Healthcare: Estimating the effectiveness of a new treatment.
  3. Polling: Predicting election outcomes based on survey results.

Conclusion

Confidence intervals provide a robust method for estimating population parameters with a quantifiable level of uncertainty. Mastering CIs enables better decision-making and clearer communication of statistical findings.

Next Steps: Hypothesis Testing