Estimating Parameters and Determining Sample Sizes: Estimating a Population Mean

Estimating Parameters and Determining Sample Sizes

Estimating a Population Mean

This section introduces methods for using a sample mean to make inferences about the value of the corresponding population mean. The focus is on estimation techniques and determining appropriate sample sizes for statistical inference.

• 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 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 level of precision.

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

Objective

To construct a confidence interval used to estimate a population mean when the population standard deviation 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 or both of the following conditions must be satisfied:

  • The population is normally distributed, or

  • \( n > 30 \) (Central Limit Theorem applies)

Confidence Interval Formulas

• The confidence interval for the population mean is given by: $\bar{x} - E < \mu < \bar{x} + E$ or equivalently, $\mu = \bar{x} \pm E$

• The margin of error is: $E = t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}}$ 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., 0.95 or 95%) is the probability that the confidence interval contains the true population mean.

• The confidence level is the complement of the significance level $\alpha$ (i.e., $\text{Confidence Level} = 1 - \alpha$).

Critical Value and Degrees of Freedom

• Critical Value: The value $t_{\alpha/2}$ separates the area $\alpha/2$ in the right tail of the Student t distribution.

• Degrees of Freedom: $df = n - 1$ is used to find the appropriate critical value.

Rounding Rules

• Original Data: When using original data values, round the confidence interval limits to one more decimal place than the original data.

• Summary Statistics: When using summary statistics ($n$, $\bar{x}$, $s$), round the confidence interval limits to the same number of decimal places as the sample mean.