Estimation and Confidence Intervals: Structured Study Notes

Estimation and Confidence Intervals

Introduction

Estimation and confidence intervals are central concepts in inferential statistics, allowing us to use sample data to make informed guesses about population parameters. Rather than relying on a single value, confidence intervals provide a range of plausible values, quantifying uncertainty and reliability in statistical conclusions.

Point and Interval Estimates

Definitions and Concepts

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

• Interval Estimate: A range of plausible values for a parameter, constructed by adding and subtracting a margin of error from the point estimate.

• Margin of Error: The maximum expected difference between the point estimate and the true parameter value.

• Confidence Interval: An interval estimate that includes the true parameter with a specified probability (confidence level).

• Unbiased Estimator: An estimator whose sampling distribution is centered at the true parameter value.

Properties of Point Estimators

• Center: The sampling distribution should be centered at the parameter (unbiasedness).

• Spread: A smaller spread (standard deviation) is preferred for greater precision.

Why Interval Estimates Matter

• Interval estimates reflect uncertainty and provide a range for plausible parameter values.

• They help communicate the reliability of statistical conclusions.

Confidence Intervals for a Population Proportion

Constructing a Confidence Interval

When outcomes are categorical, the population proportion p is estimated by the sample proportion \( \hat{p} \). For large samples, the sampling distribution of \( \hat{p} \) is approximately normal with mean p and standard error:

• $SE_{\hat{p}} = \sqrt{ \frac{p(1-p)}{n} }$

The confidence interval for p is:

• $\hat{p} \pm z^* \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }$

Where z* is the critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95%).

Example: Estimating Gene Mutation Prevalence

• Suppose 36 out of 200 individuals have a mutation: $\hat{p} = \frac{36}{200} = 0.18$

• Standard error: $\sqrt{ \frac{0.18 \times 0.82}{200} } = 0.027$

• 95% CI: $0.18 \pm 1.96 \times 0.027 = (0.127, 0.233)$

Conditions and Cautions

• Sample size should be large enough for normal approximation: $np \geq 15$ and $n(1-p) \geq 15$

Confidence Intervals for a Population Mean

Constructing a Confidence Interval

For a population mean, the confidence interval is based on the sample mean \( \bar{x} \) and the standard error:

• $SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (if population standard deviation \( \sigma \) is known)

• $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$ (if sample standard deviation \( s \) is used)

When \( \sigma \) is unknown, use the t distribution:

• $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

Where t* is the critical value from the t distribution with n-1 degrees of freedom.

Example: Enzyme Activity Levels

• Sample mean = 2.21, sample standard deviation = 0.39, n = 12

• Standard error: $\frac{0.39}{\sqrt{12}} = 0.11$

• 95% CI: $2.21 \pm 2.201 \times 0.11 = (1.97, 2.45)$

The t Distribution vs. Normal Distribution

• t distribution is used when population standard deviation is unknown and sample size is small.

• t distribution has heavier tails than the normal distribution, reflecting greater uncertainty.

Robustness and Assumptions

• Confidence intervals using the t distribution are robust to moderate departures from normality.

• Check for outliers and skewness; large samples mitigate non-normality effects.

Choosing a Sample Size

Sample Size for Estimating a Mean

To achieve a desired margin of error E at confidence level 1-\alpha:

• $n = \left( \frac{z^* \sigma}{E} \right)^2$

If \sigma is unknown, use an estimate from a pilot study.

Sample Size for Estimating a Proportion

• $n = \frac{z^{*2} p^* (1-p^*)}{E^2}$

If no prior estimate is available, use p* = 0.5 for the most conservative sample size.

Example: Planning a Customer Satisfaction Survey

• Desired margin of error = 0.05, confidence level = 95%, $z^* = 1.96$, $p^* = 0.5$

• $n = \frac{1.96^2 \times 0.5 \times 0.5}{0.05^2} = 384.16$

• Round up to 385 respondents.

Relationship Between Sample Size and Margin of Error

• Increasing sample size decreases margin of error, but with diminishing returns.

Key Terms Recap

Keyword	Definition
Point estimator	A single statistic that estimates a population parameter.
Interval estimator	A range of plausible values for a parameter, constructed by adding and subtracting a margin of error from the point estimate.
Margin of error	Allowed difference between the point estimate and the true parameter.
Confidence interval	An interval estimate that includes the true parameter with a specified probability.
Unbiased estimator	An estimator whose sampling distribution is centered at the parameter.
Standard error	The estimated standard deviation of the sampling distribution.
Critical value	A quantile of the standard normal or t distribution used to construct confidence intervals.
t distribution	A family of distributions used when estimating means with unknown population standard deviation.
Sample size for a mean	Number of observations needed to estimate a mean with desired margin of error.
Sample size for a proportion	Number of observations needed to estimate a proportion with desired margin of error.
Conservative estimate	When no prior estimate for p is available, use p* = 0.5 for largest sample size.
Rounding up	Always round up the computed sample size to ensure the actual margin of error is not exceeded.

Summary

• Point and interval estimates are essential for statistical inference.

• Confidence intervals quantify uncertainty and provide a range for plausible parameter values.

• Sample size calculations ensure desired precision in estimates.

• Statistical software (e.g., JMP) can assist in constructing confidence intervals and determining sample size.