7.2

Estimating Parameters and Determining Sample Sizes

Introduction

This chapter focuses on statistical methods for estimating population parameters, specifically the population mean (μ), and determining the sample size required for such estimation. The main tools discussed are point estimates, confidence intervals, and sample size calculations.

Point Estimates and Confidence Intervals

Key Concepts

• Point Estimate: The sample mean (\bar{x}) is the best single-value estimate of the population mean (μ).

• Confidence Interval: An interval estimate, based on sample data, that is likely to contain the true value of the population mean with a specified level of confidence.

• Sample Size: The number of observations required to estimate a population mean with a desired margin of error.

Confidence Interval for Estimating a Population Mean

Objective

To construct a confidence interval for the population mean (μ), using sample data.

The general form of a confidence interval for the mean is:

$\bar{x} - E < μ < \bar{x} + E$

or equivalently,

$\bar{x} \pm E$

Notation

• μ: Population mean

• \bar{x}: Sample mean

• s: Sample standard deviation

• σ: Population standard deviation

• n: Number of sample values

• E: Margin of error

Confidence Interval When σ Is Known

If the population standard deviation (σ) is known, the confidence interval is constructed using the standard normal (z) distribution.

• Margin of Error:

• $E = z_{\alpha/2} \cdot \frac{σ}{\sqrt{n}}$

• Confidence Interval:

• $\bar{x} \pm E$

Confidence Interval When σ Is Unknown

When the population standard deviation (σ) is unknown, the sample standard deviation (s) is used, and the Student's t-distribution is applied, especially for small samples (n ≤ 30).

• Requirements:

  • The sample is a simple random sample.

  • σ is unknown.

  • Either the population is normally distributed, or n > 30.

• Margin of Error:

• $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

• Degrees of Freedom: $df = n - 1$

• Confidence Interval:

• $\bar{x} - E < μ < \bar{x} + E$

Student's t-Distribution

Key Properties

• If a population is normal, the statistic $t = \frac{\bar{x} - μ}{s/\sqrt{n}}$ follows a Student's t-distribution for all sample sizes n.

• The t-distribution is symmetric and bell-shaped, like the standard normal distribution, but has heavier tails (more variability), especially for small n.

• The mean of the t-distribution is 0; the standard deviation is greater than 1 and depends on n.

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

• Degrees of Freedom (df): $df = n - 1$

• Finding Critical Values: Use statistical tables or technology. If the exact df is not available, use the closest lower df or interpolate.

Worked Example: Confidence Interval for Peanut Butter Cups

Problem Statement

Given the weights (in grams) of 6 randomly selected Reese's Peanut Butter Cups Miniatures from a package of 38 cups (total weight 340.2 g), estimate the mean weight and construct a 95% confidence interval.

• Sample data: 8.639, 8.689, 8.548, 8.980, 8.936, 9.042

• Population mean (if filled as labeled): $340.2 / 38 = 8.953$ g

Solution Steps

1. Point Estimate: $\bar{x} = 8.8057$ g

2. Requirement Check: The sample is random and appears to be from a normal population (verified by normal quantile plot).

3. Calculate Standard Deviation: $s = 0.2054$ g

4. Degrees of Freedom: $df = 6 - 1 = 5$

5. Critical Value: $t_{0.025} = 2.571$ (from t-table for 95% confidence)

6. Margin of Error:

   $E = 2.571 \cdot \frac{0.2054}{\sqrt{6}} = 0.215589$

7. Confidence Interval:

   $8.8057 - 0.215589 < μ < 8.8057 + 0.215589$

   $8.5901 < μ < 9.0213$ g

8. Interpretation: We are 95% confident that the true mean weight of the cups is between 8.5901 g and 9.0213 g. Since 8.953 g (the labeled mean) is within this interval, the packaging appears accurate.

Finding a Point Estimate and Margin of Error from a Confidence Interval

• Point Estimate of μ:

  $μ = \frac{\text{upper confidence limit} + \text{lower confidence limit}}{2}$

• Margin of Error:

  $E = \frac{\text{upper confidence limit} - \text{lower confidence limit}}{2}$

Determining Sample Size for Estimating a Population Mean

Objective

To determine the sample size (n) required to estimate the population mean (μ) with a specified margin of error (E).

Notation

• μ: Population mean

• σ: Population standard deviation

• \bar{x}: Sample mean

• E: Desired margin of error

• z_{\alpha/2}: z-score for the desired confidence level

Sample Size Formula

The required sample size is:

$n = \left[ \frac{z_{\alpha/2} \cdot σ}{E} \right]^2$

Round-Off Rule: If n is not a whole number, always round up to the next whole number.

Worked Example: IQ Scores of Statistics Students

• Desired confidence level: 95% ($z_{0.025} = 1.96$)

• Desired margin of error: $E = 3$ IQ points

• Assumed population standard deviation: $σ = 15$

Plug into the formula:

$n = \left[ \frac{1.96 \times 15}{3} \right]^2 = [9.8]^2 = 96.04$

Round up: $n = 97$

Interpretation: A sample of at least 97 statistics students is needed to be 95% confident that the sample mean is within 3 points of the true population mean.

Choosing Between t and z Distributions

• Use the z-distribution when the population standard deviation (σ) is known and/or the sample size is large (n > 30).

• Use the t-distribution when σ is unknown and the sample size is small (n ≤ 30), provided the population is approximately normal.