Estimating Parameters and Determining Sample Sizes: Confidence Intervals and Margin of Error

Estimating Parameters and Determining Sample Sizes

Introduction to Confidence Intervals

Confidence intervals are a fundamental concept in inferential statistics, allowing us to estimate population parameters based on sample data. They provide a range of plausible values for an unknown parameter, such as a population mean or proportion, with a specified level of confidence.

• Confidence Interval (CI): An interval estimate, derived from sample statistics, that is likely to contain the true value of a population parameter.

• Confidence Level: The probability (expressed as a percentage, e.g., 90%, 95%) that the confidence interval actually contains the population parameter.

• Margin of Error (E): The maximum expected difference between the point estimate and the true population parameter.

Example: If a poll reports that 26% of professionals cite a specific interview turnoff with a margin of error of ±3 percentage points, the confidence interval is 23% to 29%.

Key Terms and Definitions

• Point Estimate: A single value estimate of a population parameter (e.g., sample mean \( \bar{x} \) or sample proportion \( \hat{p} \)).

• Sample Proportion (\( \hat{p} \)): The proportion of successes in the sample.

• Sample Size (n): The number of observations in the sample.

• Complement of Sample Proportion (\( \hat{q} \)): Calculated as \( 1 - \hat{p} \).

• Population Proportion (p): The true proportion in the population (usually unknown).

• Critical Value (\( z_{\alpha/2} \)): The z-score that corresponds to the desired confidence level.

• Significance Level (\( \alpha \)): The probability that the confidence interval does not contain the population parameter (\( \alpha = 1 - \text{confidence level} \)).

Constructing Confidence Intervals for Proportions

To estimate a population proportion, use the following formula for the confidence interval:

• \( \hat{p} \pm E \)

• Where the margin of error \( E \) is given by:

$E = z_{\alpha/2} \sqrt{ \dfrac{\hat{p}\hat{q}}{n} }$

• \( z_{\alpha/2} \): Critical value from the standard normal distribution for the desired confidence level.

• \( \hat{p} \): Sample proportion.

• \( \hat{q} = 1 - \hat{p} \)

• \( n \): Sample size.

Example: If 12% of 500 respondents prefer chocolate pie, and the margin of error is ±5%, the confidence interval is 7% to 17%.

Interpreting Confidence Intervals

• A 95% confidence interval means that if the same population is sampled repeatedly, approximately 95% of the calculated intervals will contain the true population parameter.

• The confidence level does not indicate the probability that the specific interval calculated from a single sample contains the parameter, but rather the reliability of the estimation process.

Example: "We are 95% confident that the true proportion of people who prefer chocolate pie is between 7% and 17%."

Critical Values and Confidence Levels

The critical value \( z_{\alpha/2} \) depends on the confidence level:

Confidence Level	\( \alpha \)	\( z_{\alpha/2} \)
80%	0.20	1.28
85%	0.15	1.44
90%	0.10	1.645
94%	0.06	1.88
95%	0.05	1.96
99%	0.01	2.576

Additional info: Table values inferred from standard normal distribution tables.

Expressing Confidence Intervals

• Confidence intervals can be written as \( \hat{p} \pm E \) or as lower and upper bounds: \( \hat{p} - E < p < \hat{p} + E \).

• Example: If \( 0.666 < p < 0.888 \), then \( \hat{p} = 0.777 \) and \( E = 0.111 \).

Comparing Confidence Intervals

• Higher confidence levels (e.g., 99%) produce wider intervals than lower confidence levels (e.g., 90%) for the same data, because more certainty requires a broader range.

• Wider intervals are less precise but more likely to contain the true parameter.

Sample Size Determination

To achieve a desired margin of error for a given confidence level, the required sample size can be calculated:

• For proportions:

$n = \left( \dfrac{z_{\alpha/2}^2 \cdot \hat{p} \hat{q}}{E^2} \right)$

• If no prior estimate for \( \hat{p} \) is available, use \( \hat{p} = 0.5 \) for maximum variability.

• Smaller margins of error or higher confidence levels require larger sample sizes.

Example: To estimate the proportion of adults who gamble online with a margin of error of 5% and 90% confidence, the required sample size is calculated using the formula above.

Confidence Intervals for Means

When estimating a population mean, the confidence interval is:

$\bar{x} \pm t_{\alpha/2, df} \left( \dfrac{s}{\sqrt{n}} \right)$

• \( \bar{x} \): Sample mean

• \( t_{\alpha/2, df} \): Critical value from the t-distribution with \( df = n-1 \) degrees of freedom

• \( s \): Sample standard deviation

• \( n \): Sample size

Example: For a sample mean of 19.76 minutes, margin of error 1.301 minutes, the 95% confidence interval is 18.46 < y < 21.06 minutes.

Requirements and Interpretation

• For means, the sample should be random, and either the population is normally distributed or the sample size is large (n ≥ 30).

• Interpretation: "We are 95% confident that the true mean lies between the lower and upper bounds of the interval."

Comparing Two Confidence Intervals

• If two confidence intervals for different groups (e.g., men and women) overlap, there is no strong evidence of a difference between the groups.

• If the intervals do not overlap, this suggests a statistically significant difference.

Summary Table: Key Elements in Confidence Interval Estimation

Symbol	Meaning
\( \hat{p} \)	Sample proportion
\( \hat{q} \)	1 - \( \hat{p} \) (complement of sample proportion)
n	Sample size
E	Margin of error
p	Population proportion
\( z_{\alpha/2} \)	Critical value for confidence level
\( \alpha \)	Significance level (1 - confidence level)

Applications and Examples

• Estimating the proportion of survey respondents with a certain preference.

• Determining if a sample provides evidence against a hypothesized population value (e.g., proportion of boys in births, percentage of yellow peas in genetics experiments).

• Calculating the required sample size for a desired margin of error and confidence level in various real-world contexts (e.g., airline passenger preferences, online gambling rates).

Conclusion

Understanding confidence intervals and sample size determination is essential for making reliable inferences about populations from sample data. Proper interpretation and calculation ensure that statistical conclusions are both accurate and meaningful in practical applications.