Confidence Interval Estimation

Introduction

Confidence interval estimation is a fundamental concept in statistics for business, providing a range of plausible values for unknown population parameters based on sample data. This chapter focuses on constructing and interpreting confidence intervals for the population mean and proportion, and determining the required sample size for desired precision.

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 mean or proportion.

Point and Interval Estimates

Definitions and Concepts

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

• Interval Estimate (Confidence Interval): A range of values, derived from sample statistics, that is likely to contain the population parameter. The width of the interval reflects the margin of error (MOE).

Table: Point Estimates

Confidence Intervals

Purpose and Interpretation

• Quantifies the uncertainty associated with a point estimate.

• Provides more information about a population characteristic than a point estimate alone.

• Expressed in terms of a confidence level (e.g., 95%, 99%).

• Can never be 100% confident; some uncertainty always remains.

General Formula

The general formula for all confidence intervals is:

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

• Critical Value: A value from the sampling distribution (Z or t) corresponding to the desired confidence level.

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

• Margin of Error (MOE): The product of the critical value and the standard error.

Confidence Level

• The probability that the interval contains the unknown population parameter.

• Common levels: 90%, 95%, 99%.

• Expressed as (e.g., for 95% confidence).

• Interpretation: If many samples are taken, approximately (1 - )100% of the intervals will contain the true parameter.

Confidence Interval for the Population Mean

Case 1: Population Standard Deviation () Known

• Assumptions:

  • Population standard deviation () is known.

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

• Formula:

  • : sample mean

  • : critical value from standard normal distribution

  • : standard error

Common Z Critical Values

Example

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

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

Case 2: Population Standard Deviation () Unknown

• Assumptions:

  • Population standard deviation is unknown.

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

• Use Student's t Distribution:

  • depends on degrees of freedom ().

• Formula:

  • : sample standard deviation

  • : critical value from t distribution

Degrees of Freedom (df)

• Number of independent values in the sample after calculating the mean: .

Selected t Distribution Values

Example

A random sample of has and . For 95% confidence, , :

Checking Normality

• If normality is not stated, check using a normal probability plot or boxplot.

Confidence Interval for the Population Proportion

Concepts and Formula

• Used to estimate the true proportion () in the population.

• Sample proportion () is the point estimate.

• Standard error of the sample proportion:

Estimated as

• Confidence Interval Formula:

• Requirements: and for normal approximation.

Example

A sample of 100 people shows 25 are left-handed (). For 95% confidence:

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

Effect of Sample Size and Confidence Level on Interval Width

• Increasing sample size () decreases the margin of error and narrows the interval.

• Increasing confidence level increases the margin of error and widens the interval.

• Decreasing confidence level narrows the interval.

Determining Required Sample Size

For Mean

• To achieve a desired margin of error () at a specified confidence level:

• If is unknown, use a pilot sample to estimate with .

Example

If , desired , and 90% confidence ():

For Proportion

• To achieve a desired margin of error () at a specified confidence level:

• If is unknown, use a pilot sample or conservatively use .

Example

Estimate the true proportion of sales invoices containing errors within at 95% confidence (), pilot sample :

(round up to 100)

Conclusion

• Always report a confidence interval estimate alongside a point estimate.

• Include the confidence level and sample size in reporting.

• Provide an interpretation of the confidence interval in context.

Chapter Summary

• Construction and interpretation of confidence intervals for mean and proportion.

• Determination of required sample size for desired precision in confidence intervals.