Confidence Interval Estimation – Business Statistics Chapter 8 Study Notes

Confidence Interval Estimation

Introduction

Confidence interval estimation is a fundamental concept in business statistics, providing a range of values within which a population parameter is likely to fall. This chapter covers the construction and interpretation of confidence intervals for means and proportions, and explains how to determine the necessary sample size for reliable estimation.

Learning Objectives

• To construct and interpret confidence interval estimates for the mean and the proportion.

• To determine the sample size necessary to develop a confidence interval for the population mean or population proportion.

Point and Interval Estimates

Definitions and Concepts

• Point Estimate: A single value used to estimate a population parameter (e.g., sample mean ̅X for population mean μ).

• Confidence Interval: A range of values, derived from sample statistics, that is likely to contain the population parameter. It reflects the variability or uncertainty of the estimate.

• Margin of Error (e): The amount added and subtracted from the point estimate to create the confidence interval.

Example Table:

Population Parameter	Sample Statistic (Point Estimate)
Mean (μ)	Sample Mean (̅X)
Proportion (π)	Sample Proportion (p)

General Formula for Confidence Interval

The general formula for a confidence interval is:

$ext{Point Estimate} \\pm ( ext{Critical Value}) imes ( ext{Standard Error})$

• Point Estimate: The sample statistic estimating the population parameter.

• Critical Value: Based on the sampling distribution and desired confidence level (e.g., Z or t value).

• Standard Error: The standard deviation of the sampling distribution of the point estimate.

Confidence Intervals for the Population Mean

When Population Standard Deviation (σ) is Known

• Assumptions:

  • Population standard deviation σ is known.

  • Population is normally distributed, or sample size is large (n > 30).

• Formula: $\bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ where $\bar{X}$ is the sample mean, $Z_{\alpha/2}$ is the critical value from the standard normal distribution, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Finding the Critical Value, Z

• For a 95% confidence interval, $Z = \pm 1.96$.

• Common confidence levels and Z values:

Confidence Level	Z Value
90%	1.645
95%	1.96
99%	2.58

Example

• Sample mean resistance: 2.20 ohms

• Population standard deviation: 0.35 ohms

• Sample size: 11

• 95% confidence interval: $2.20 \pm 1.96 \times \frac{0.35}{\sqrt{11}} = 2.20 \pm 0.2068$ $1.9932 \leq \mu \leq 2.4068$

Interpretation

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

• 95% of intervals constructed in this manner will contain the true mean.

Using Excel for Confidence Intervals

• Functions: NORMINV(probability, mean, standard deviation)

• For 95% CI: Lower = NORMINV(0.025, 2.2, 0.1055) = 1.9932 Upper = NORMINV(0.975, 2.2, 0.1055) = 2.4068

• For 99% CI: Lower = NORMINV(0.005, 2.2, 0.1055) = 1.9282 Upper = NORMINV(0.995, 2.2, 0.1055) = 2.4718

Determining Desired Sample Size for Estimating the Mean

• Let e = desired margin of error.

• Formula: $e = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ Solve for n: $n = \frac{Z_{\alpha/2}^2 \sigma^2}{e^2}$

• Example: If $\sigma = 45$, $e = 5$, $Z = 1.645$ (for 90% confidence): $n = \frac{(1.645)^2 \times 45^2}{5^2} = 219.19$ Required sample size: $n = 220$ (always round up)

If σ is Unknown

• Estimate σ using a pilot sample or sample standard deviation S.

Confidence Interval for the Mean (σ Unknown, Small Sample)

Student's t Distribution

• Use when population standard deviation is unknown and sample size is small.

• 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.

• As n increases, t distribution approaches the normal (Z) distribution.

Confidence Level	t (10 d.f.)	t (20 d.f.)	t (30 d.f.)	Z
0.80	1.372	1.325	1.310	1.28
0.90	1.812	1.725	1.697	1.645
0.95	2.228	2.086	2.042	1.96
0.99	3.169	2.845	2.750	2.58

Example

• Sample mean: 50, Sample standard deviation: 8, Sample size: 25

• Degrees of freedom: 24, $t_{0.025,24} = 2.064$

• 95% CI: $50 \pm 2.064 \times \frac{8}{\sqrt{25}} = 50 \pm 3.302$ $46.698 \leq \mu \leq 53.302$

Comparison with Z Distribution

• Small differences in interval width between t and Z distributions, especially for small samples.

• Increasing confidence level increases interval width; increasing sample size decreases interval width.

Alternative Approach: Descriptive Statistics from Data Analysis

• If individual data points are available, use descriptive statistics to compute mean and standard deviation, then construct the confidence interval.

• If only summary statistics are available, use t distribution or related functions.

Confidence Interval for the Difference Between Two Means

Related (Paired) Samples

• Each observation in one sample is paired with an observation in the other sample.

• Example: Twin study comparing boys and girls' responses to an advertisement.

• Construct CI for the mean difference as for a single variable.

Unrelated (Independent) Samples

• Samples are independent; use pooled variance for CI calculation.

• Formula for pooled variance: $S_p^2 = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1 + n_2 - 2}$

• CI for difference in means: $(\bar{X}_1 - \bar{X}_2) \pm t_{\alpha/2} \sqrt{S_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$

Confidence Intervals for the Population Proportion

Formula and Assumptions

• For large samples ($np \geq 15$ and $n(1-p) \geq 15$), the sampling distribution of the sample proportion is approximately normal.

• Formula: $p \pm Z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$

Example

• Sample: 100 people, 25 left-handed ($p = 0.25$)

• 95% CI: $0.25 \pm 1.96 \times \sqrt{\frac{0.25 \times 0.75}{100}} = 0.25 \pm 0.0849$ $0.1651 \leq \pi \leq 0.3349$

Using Excel

• Lower Limit: NORMINV(0.025, 0.25, 0.0433) = 0.1651

• Upper Limit: NORMINV(0.975, 0.25, 0.0433) = 0.3349

Determining Desired Sample Size for Proportion

• Formula: $n = \frac{Z_{\alpha/2}^2 p(1-p)}{e^2}$

• Where e is the desired margin of error, p is the estimated proportion (use pilot sample or 0.5 for maximum variability).

Ethical Issues in Reporting Confidence Intervals

• Always report the confidence interval, confidence level, sample size, and interpretation when presenting point estimates.

Summary Table: Confidence Interval Formulas

Parameter	Known/Unknown	Formula
Mean (μ)	σ known	$\bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
Mean (μ)	σ unknown	$\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$
Proportion (π)	Large sample	$p \pm Z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$