Estimation: Single Population – Confidence Intervals and Point Estimation

Chapter 7: Estimation – Single Population

Chapter Overview

This chapter introduces the concepts of point estimation and confidence intervals for single population parameters, including the mean, proportion, and variance. It covers the construction and interpretation of confidence intervals using both the Z and t distributions, and discusses the properties of estimators such as unbiasedness and efficiency.

Section 7.1: Properties of Point Estimators

Definition of Estimator and Estimate

• Estimator: A random variable (function of sample data) used to approximate an unknown population parameter.

• Estimate: The specific value obtained from an estimator after observing the sample.

Example: To estimate the mean delivery time for an online company, the sample mean (estimator) is calculated from observed delivery times, and the resulting number is the estimate.

Estimator vs. Estimate

• The estimator is a process (e.g., sample mean or median).

• The estimate is the result (a single number) from applying the estimator to data.

Example: Estimating weekly sales of orange juice: the sample mean is an estimator; the calculated value is the estimate.

Point and Interval Estimates

• Point Estimate: A single value used to estimate a population parameter.

• Confidence Interval: A range of values that likely contains the population parameter, reflecting sampling variability.

Diagram: Lower Confidence Limit – Point Estimate – Upper Confidence Limit (Width = confidence interval width)

Table: Point Estimates

Parameter	Estimator (random variable)	Sample Statistic (point estimate)
Mean: μ	\( \bar{X} \)	\( \bar{x} \)
Proportion: p	\( \hat{p} \)	\( \hat{p} \)

Properties of Estimators

• Unbiasedness: An estimator is unbiased if its expected value equals the parameter it estimates: $E(\hat{\theta}) = \theta$

• Examples:

  • Sample mean (\( \bar{X} \)) is an unbiased estimator of μ.

  • Sample variance (\( S^2 \)) is an unbiased estimator of σ2.

  • Sample proportion (\( \hat{p} \)) is an unbiased estimator of p.

• Bias: The bias of an estimator \( \hat{\theta} \) is $\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta$

• Most Efficient Estimator: Among unbiased estimators, the one with the smallest variance is most efficient (minimum variance unbiased estimator).

• Relative Efficiency: $\text{Relative Efficiency} = \frac{\text{Var}(\hat{\theta}_2)}{\text{Var}(\hat{\theta}_1)}$

Table: Properties of Common Point Estimators

Population Parameter	Point Estimator	Properties
Mean, μ	\( \bar{X} \)	Unbiased, most efficient (assuming normality)
Median, μ1/2	Median	Unbiased (assuming normality), not most efficient
Proportion, p	\( \hat{p} \)	Unbiased, most efficient
Variance, σ2	\( S^2 \)	Unbiased, most efficient (assuming normality)

Additional info: The median estimator is typically used for the median, but can also be unbiased for the mean under certain conditions.

Confidence Interval Estimation

Concept and Interpretation

• A confidence interval provides a range of plausible values for a population parameter, reflecting uncertainty due to sampling variability.

• The confidence level (e.g., 95%) indicates the proportion of such intervals that would contain the true parameter in repeated sampling.

• General form: $\hat{\theta} \pm ME$ (Point Estimate ± Margin of Error)

Confidence Interval and Confidence Level

• If $P(a < \theta < b) = 1 - \alpha$, then [a, b] is a 100(1-α)% confidence interval for θ.

• Common confidence levels: 90%, 95%, 98%, 99%.

• For a specific interval, the true parameter either is or is not contained; the confidence level refers to the method, not a single interval.

General Formula for Confidence Intervals

• All confidence intervals can be written as: $\hat{\theta} \pm ME$

• Where ME (margin of error) depends on the desired confidence level and sample size.

Section 7.2: Confidence Interval Estimation for the Mean (σ2 Known)

Assumptions

• Population variance σ2 is known.

• Population is normally distributed (or large sample size by Central Limit Theorem).

Confidence Interval Formula

• $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

• Where $z_{\alpha/2}$ is the standard normal value for the desired confidence level.

Endpoints

• Upper Confidence Limit (UCL): $\bar{x} + z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

• Lower Confidence Limit (LCL): $\bar{x} - z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Margin of Error

• $ME = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

• Interval width $w = 2 \times ME$

• Reducing ME: decrease σ, increase n, or decrease confidence level (1-α).

Common Z Values for Confidence Levels

Confidence Level	Confidence Coefficient (1-α)	$z_{\alpha/2}$ value
80%	0.80	1.28
90%	0.90	1.645
95%	0.95	1.96
98%	0.98	2.33
99%	0.99	2.58

Example

A sample of 11 circuits from a normal population has a sample mean resistance of 2.20 ohms and known population standard deviation of 0.35 ohms. The 95% confidence interval for the true mean resistance is:

$2.20 \pm 1.96 \left(\frac{0.35}{\sqrt{11}}\right) = (1.9932, 2.4068)$

Interpretation: We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms.

Section 7.3: Confidence Interval Estimation for the Mean (σ2 Unknown)

Student's t Distribution

• Used when population standard deviation is unknown.

• Assumes population is normally distributed.

• Degrees of freedom: $df = n - 1$

• As n increases, t-distribution approaches the standard normal distribution.

$T = \frac{\bar{X} - \mu}{s/\sqrt{n}} \sim t(n-1)$

Confidence Interval Formula

• $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$

• Where $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with n-1 degrees of freedom.

Example: For n = 25, $\bar{x} = 50$, $s = 8$, 95% CI:

$50 \pm 2.0639 \left(\frac{8}{\sqrt{25}}\right) = (46.698, 53.302)$

Interpretation: We are 95% confident that the true mean is between 46.698 and 53.302.

Section 7.4: Confidence Interval Estimation for Population Proportion

Confidence Interval for Proportion

• For large samples, the sampling distribution of the sample proportion is approximately normal.

• Standard error: $\sqrt{\frac{p(1-p)}{n}}$ (estimated with sample data)

• Confidence interval: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Example: In a sample of 100 people, 25 are left-handed. 95% CI for the true proportion:

$0.25 \pm 1.96 \sqrt{\frac{0.25 \times 0.75}{100}} = (0.1651, 0.3349)$

Interpretation: We are 95% confident that the true proportion of left-handers is between 16.51% and 33.49%.

Section 7.5: Confidence Interval Estimation for the Variance

Confidence Interval for Population Variance

• Assumes population is normally distributed.

• Based on the chi-square distribution with (n-1) degrees of freedom.

• Confidence interval for variance $\sigma^2$:

$\left( \frac{(n-1) s^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1) s^2}{\chi^2_{1-\alpha/2, n-1}} \right)$

Example: For n = 17, $s^2 = 74$, 95% CI for variance:

$\left( \frac{16 \times 74}{27.587}, \frac{16 \times 74}{6.908} \right) = (43, 171)$ (values are illustrative)

For standard deviation, take square roots of the endpoints.

Summary

• Introduced point and interval estimation.

• Developed confidence intervals for mean (known and unknown variance), proportion, and variance.

• Discussed properties of estimators: unbiasedness and efficiency.