Estimating a Population Mean and Confidence Intervals

Estimating a Population Mean

Introduction

Estimating a population mean is a fundamental concept in inferential statistics. It involves using sample data to make inferences about the mean of a larger population. Confidence intervals provide a range of plausible values for the population mean, reflecting the uncertainty inherent in sampling.

Point Estimate of the Population Mean

• Sample Mean (𝑛): The best point estimate of the population mean (μ) is the sample mean.

• Point estimation uses sample statistics to estimate population parameters.

Confidence Interval for Population Mean (μ)

A confidence interval gives a range of values within which the population mean is likely to fall, based on sample data. The method used depends on whether the population standard deviation (σ) is known or unknown.

Method 1: σ Unknown (Use t-distribution)

• Most common scenario: population standard deviation is unknown.

• Use the Student t-distribution.

Formula for Margin of Error (E):

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

• E: Margin of Error

• t𝑛/2: Critical value from t-distribution

• s: Sample standard deviation

• n: Sample size

• df: Degrees of freedom, $df = n - 1$

Confidence Interval:

$\bar{x} - E < \mu < \bar{x} + E$

Steps to Construct a Confidence Interval for μ (when σ unknown)

1. Verify requirements: sample must be random, population normally distributed or $n > 30$.

2. For desired confidence level (e.g., 90%, 95%, 99%), find α.

3. Find $t_{𝑛/2}$ using calculator or t-table.

4. Compute E and construct the interval.

Rounding Rules

• Round the confidence interval to one decimal place beyond the data values.

• If data values are given, round the interval to the same number of decimal places as the data.

Understanding the t-Distribution

• Used when population standard deviation is unknown and sample size is small ($n < 30$).

• As sample size increases, the t-distribution approaches the normal distribution.

• t-scores are calculated as:

  $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

• Degrees of freedom: $df = n - 1$

Calculator Instructions (TI-83/84)

• Use TInterval function under Stats to compute confidence intervals.

• Input sample mean, standard deviation, sample size, and confidence level.

Example 1: Verizon Airport Wifi Data Speeds

• Sample mean: 17.6 Mbps

• Sample standard deviation: 6.01713 Mbps

• Sample size: 50

• 95% confidence interval calculated as:

  $E = t_{0.025} \cdot \frac{6.01713}{\sqrt{50}} = 2.0096 \cdot 0.8514 = 1.709$

  Interval: $17.6 - 1.709 < \mu < 17.6 + 1.709$

  We are 95% confident that the population mean is between 15.89 and 19.31 Mbps.

Example 2: DNA Sample Data

• Nominal data (letters) cannot be used for mean or confidence interval calculations.

• For numerical data, use TInterval as above.

Example 3: Heart Rates

• Compare 95% confidence intervals for male and female pulse rates.

• Male: (67.8, 71.4) bpm

• Female: (72.0, 76.1) bpm

• Intervals do not overlap, suggesting a difference in mean heart rates.

Determining Sample Size for Estimating μ

To estimate a population mean with a specified margin of error, use:

$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$

• n: Required sample size

• z𝑛/2: Critical value from normal distribution

• σ: Population standard deviation

• E: Desired margin of error

Estimating σ:

• Use range rule of thumb: $\sigma \approx \frac{\text{Range}}{4}$

• Use sample standard deviation if available

• Use previous study results

Example 4: IQ Test Sample Size

• Mean IQ = 100, σ = 15

• 95% confidence, E = 2

• $n = \left( \frac{1.96 \cdot 15}{2} \right)^2 = 216.09$

• Round up to 217

Method 2: σ Known (Use z-distribution)

• Rare in practice; use when population standard deviation is known.

• Formula for margin of error:

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

Confidence Interval:

$\bar{x} - E < \mu < \bar{x} + E$

Example 5: Birth Weights

• Sample mean: 3.6 kg

• Sample size: 205

• Population standard deviation: 0.7 kg

• 95% confidence interval:

• $E = 1.96 \cdot \frac{0.7}{\sqrt{205}} = 0.096$

• Interval: $3.6 - 0.096 < \mu < 3.6 + 0.096$

Choosing Between t and z Distributions

The choice of distribution depends on sample size and knowledge of population standard deviation.

Condition	Method
σ not known and normally distributed population (or n > 30)	Use Student t distribution
σ known and normally distributed population (or n > 30)	Use normal (z) distribution
Population is not normally distributed and n is small	Use nonparametric methods

Key Points and Summary

• Confidence intervals provide a range for the population mean based on sample data.

• Use t-distribution when σ is unknown and sample size is small.

• Use z-distribution when σ is known or sample size is large.

• Sample size calculations help ensure desired precision in estimates.

• Interpretation of intervals should consider overlap and context.

Additional info: The notes also emphasize the importance of proper interpretation of confidence intervals and the limitations when dealing with nominal data.