Estimating Parameters and Determining Sample Sizes: Confidence Intervals for Population Mean

Estimating Parameters and Determining Sample Sizes

Overview

This chapter focuses on statistical methods for estimating population parameters, specifically the population mean, and determining the required sample sizes for reliable estimation. The main concepts include point estimation, confidence intervals, and sample size determination.

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

• Confidence Interval: A range of values, derived from sample data, that is likely to contain the true population mean with a specified level of confidence.

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

Confidence Interval for Estimating a Population Mean (\(\sigma\) Not Known)

Objective

To construct a confidence interval for the population mean when the population standard deviation (\sigma) is unknown.

Notation

• \(\mu\): Population mean

• \(\bar{x}\): Sample mean

• \(s\): Sample standard deviation

• \(n\): Number of sample values

• \(E\): Margin of error

Requirements

• The sample must be a simple random sample.

• Either the population is normally distributed or the sample size is large (n > 30).

Confidence Interval Formats

• \(\bar{x} - E < \mu < \bar{x} + E\)

• \(\bar{x} \pm E\)

• \((\bar{x} - E, \bar{x} + E)\)

Margin of Error Formula

The margin of error for the confidence interval is:

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

where $t_{\alpha/2}$ is the critical value from the Student t distribution with $df = n - 1$ degrees of freedom.

Confidence Level

• The confidence level (e.g., 95%) is the probability that the interval contains the true population mean.

• $\alpha$ is the complement of the confidence level (e.g., $\alpha = 0.05$ for 95% confidence).

Critical Value and Degrees of Freedom

• Critical Value ($t_{\alpha/2}$): Separates an area of $\alpha/2$ in the right tail of the Student t distribution.

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

Key Points about the Student t Distribution

Properties

• If a population is normal, the statistic $t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$ follows a Student t distribution.

• The t distribution varies with sample size (different for each $n$).

• It has a symmetric bell shape, similar to the normal distribution, but with heavier tails (more variability).

• Mean of t distribution is 0; standard deviation is greater than 1 and decreases as $n$ increases.

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

Finding Critical Values

• Use technology or t tables (e.g., Table A-3) to find $t_{\alpha/2}$ for the desired confidence level and degrees of freedom.

• If the exact df is not available in the table, use the closest or next lower df, or interpolate.

Procedure for Constructing a Confidence Interval for \(\mu\)

1. Verify requirements: simple random sample and normal population or $n > 30$.

2. With $\sigma$ unknown, use $df = n - 1$ and find $t_{\alpha/2}$ for the desired confidence level.

3. Calculate margin of error: $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

4. Construct the confidence interval using one of the formats above.

5. Round interval limits appropriately: one more decimal place than the original data, or same as sample mean if using summary statistics.

Example: Finding a Critical Value $t_{\alpha/2}$

• Given $n = 6$, $df = 5$.

• For 95% confidence, $\alpha = 0.05$, so each tail has area 0.025.

• Using technology or Table A-3, $t_{0.025} = 2.571$ for $df = 5$.

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

• Point Estimate of $\mu$: $\mu = \frac{\text{upper confidence limit} + \text{lower confidence limit}}{2}$

• Margin of Error: $E = \frac{\text{upper confidence limit} - \text{lower confidence limit}}{2}$

Estimating a Population Mean When $\sigma$ Is Known

If the population standard deviation $\sigma$ is known, use the standard normal (z) distribution:

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

Choosing the Correct Distribution

Conditions	Method
$\sigma$ not known and normally distributed population or $\sigma$ not known and $n > 30$	Use Student t distribution with $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
$\sigma$ known and normally distributed population or $\sigma$ known and $n > 30$ (rarely known)	Use normal (z) distribution with $E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

Finding the Sample Size Required to Estimate a Population Mean

Objective

Determine the sample size $n$ required to estimate the population mean $\mu$ within a specified margin of error $E$.

Notation

• $\mu$ = population mean

• $\sigma$ = population standard deviation

• $\bar{x}$ = sample mean

• $E$ = desired margin of error

• $z_{\alpha/2}$ = z score for the desired confidence level

Requirement

• The sample must be a simple random sample.

Sample Size Formula

The required sample size is:

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

Dealing with Unknown $\sigma$ When Finding Sample Size

• Range Rule of Thumb: Estimate $\sigma$ as $\text{range}/4$ from sample data.

• Start and Improve: Begin sampling, use initial $s$ as an estimate for $\sigma$, and refine as more data are collected.

• Use Prior Results: Use $\sigma$ from previous studies. If uncertain, err on the side of a larger sample size.

Example: IQ Scores of Statistics Students

• Goal: Estimate mean IQ score for statistics students with 95% confidence, margin of error $E = 3$ IQ points.

• Assume $\sigma = 15$ (from prior data).

• For 95% confidence, $z_{\alpha/2} = 1.96$.

• Sample size: $n = \left[ \frac{1.96 \times 15}{3} \right]^2 = 97$ (rounded up).

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

Additional info: These procedures are foundational for inferential statistics, allowing researchers to make reliable statements about population parameters based on sample data.