9.2 Estimating a Population Mean and Confidence Intervals

Estimating a Population Mean

Objective 1: Obtain a Point Estimate for the Population Mean

Estimating a population mean is a fundamental task in inferential statistics. A point estimate provides a single value as an estimate of a population parameter, such as the mean.

• Point Estimate: The value of a sample statistic (e.g., sample mean) used to estimate a population parameter.

• Example: Suppose we have the following sample data for miles per gallon (mpg) of 2019 Honda Accord Hybrid vehicles:

  Sample	mpg
  1	44.4
  2	44.2
  3	44.0
  4	43.9
  5	43.8
  6	43.7
  7	43.6
  8	43.5
  9	43.4
  10	43.3
  11	43.2
  12	43.1
  13	43.0
  14	42.9
  15	42.8
  16	42.7

  The sample mean ($\bar{x}$) is the point estimate for the population mean ($\mu$).

Objective 2: State Properties of Student's t-Distribution

The Student's t-distribution is used when estimating population parameters when the sample size is small and/or the population standard deviation is unknown.

• Definition: The t-distribution is a family of distributions that are symmetric and bell-shaped, but have heavier tails than the normal distribution.

• Formula: The t-statistic is calculated as: $t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$

• Properties of the t-distribution:

  1. It is symmetric about zero.

  2. It has heavier tails than the standard normal distribution, especially for small sample sizes.

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

  4. The exact shape depends on the degrees of freedom ($df = n - 1$).

• Application: Used for constructing confidence intervals and hypothesis tests for means when population standard deviation is unknown.

• Example: If a sample of size $n = 10$ is drawn from a normal population, the t-distribution with $df = 9$ is used.

Objective 3: Determine t-Values

t-values are critical values from the t-distribution used in confidence interval estimation and hypothesis testing.

• Definition: A t-value corresponds to a specified probability in the tails of the t-distribution, depending on degrees of freedom.

• Finding t-values: Use t-tables to find the t-value for a given confidence level and degrees of freedom. Example: For a right-tail area of 0.01 and $df = 10$, find $t_{0.01,10}$ using the t-table.

• If the exact degrees of freedom are not listed, use the closest available value in the table.

Objective 4: Construct and Interpret a Confidence Interval for a Population Mean

A confidence interval provides a range of values within which the population mean is likely to fall, with a specified level of confidence.

• Conditions for constructing a confidence interval:

  1. The sample is a simple random sample.

  2. The population from which the sample is drawn is normally distributed, or the sample size is large ($n \geq 30$).

  3. The population standard deviation is unknown.

• Formula for confidence interval: Lower bound: $\bar{x} - t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ Upper bound: $\bar{x} + t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ where $t_{\alpha/2}$ is the critical value from the t-distribution.

• Interpretation: A 95% confidence interval means that if we were to take many samples and construct intervals, about 95% of them would contain the true population mean.

• Example: Using the sample mpg data, construct a 95% confidence interval for the mean mpg of 2019 Honda Accord Hybrid vehicles.

Objective 5: Determine the Sample Size Necessary to Estimate a Population Mean Within a Specified Margin of Error

Determining the required sample size ensures that the estimate of the population mean meets a desired level of precision.

• Margin of Error Formula: $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

• Sample Size Formula: $n = \left( \frac{t_{\alpha/2} \cdot s}{E} \right)^2$

• Example: To estimate the mean mpg within 0.5 mpg with 95% confidence, use the above formula to calculate the required sample size.

Summary Table: Key Formulas for Estimating a Population Mean

Concept	Formula
Point Estimate	$\bar{x}$
t-Statistic	$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$
Confidence Interval	$\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
Margin of Error	$E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
Sample Size	$n = \left( \frac{t_{\alpha/2} \cdot s}{E} \right)^2$

Additional info: These notes expand on the provided examples and objectives, adding definitions, formulas, and context for clarity and completeness.