Estimating a Population Mean: Confidence Intervals and Levels

Estimating a Population Mean

Branches of Statistics

Statistics is divided into two main branches: Descriptive Statistics and Inferential Statistics.

• Descriptive Statistics: Involves summarizing sample data using graphs and sample statistics. Commonly used in initial chapters to describe data sets.

• Inferential Statistics: Uses sample statistics and confidence intervals to estimate population parameters, such as proportion and mean.

Point Estimates vs. Interval Estimates

When estimating population parameters, two types of estimates are used:

• Point Estimate: A single value calculated from sample data to estimate a population parameter. For example, the sample mean ($\bar{x}$) is a point estimate of the population mean ($\mu$).

• Interval Estimate: A range of values, called a confidence interval, constructed from sample data that is likely to contain the population parameter with a specified probability (the confidence level).

Example: If the sample mean is 30.9 and the confidence interval is (27.0, 33.0), then we estimate the population mean to be between 27.0 and 33.0 with a certain confidence level.

Confidence Intervals and Levels

Definitions

• Confidence Interval (CI): A range (or interval) of values used to estimate the true value of a population parameter.

• Confidence Level: The probability that the confidence interval actually contains the population parameter. It is calculated as $1 - \alpha$, where $\alpha$ is the significance level.

Common Confidence Levels

Most Common Confidence Levels	Corresponding Values of $\alpha$
95% (0.95) confidence level	$\alpha = 0.05$
90% (0.90) confidence level	$\alpha = 0.10$
99% (0.99) confidence level	$\alpha = 0.01$

Method of Constructing a Confidence Interval for a Mean

Steps for Construction

1. Identify Sample Statistics:

   • Sample Mean ($\bar{x}$)

   • Sample Standard Deviation ($s$)

   • Sample Size ($n$)

   • Confidence Level

2. Find the Critical Value:

   • For a given confidence level and sample size, use the T Table (for small samples or unknown population standard deviation) to find the critical value $t_{\alpha/2}$.

   • The critical value depends on the confidence level and degrees of freedom ($df = n - 1$).

3. Calculate the Margin of Error ($E$):

   • Use the error formula for means: $E = t_{\alpha/2} \times \dfrac{s}{\sqrt{n}}$ where $s$ is the sample standard deviation and $n$ is the sample size.

4. Construct the Confidence Interval:

   • Lower Bound: $\bar{x} - E$

   • Upper Bound: $\bar{x} + E$

   • Expressed as: $\bar{x} - E < \mu < \bar{x} + E$

5. Interpret the Interval:

   • State that you are X% confident that the population mean $\mu$ is within the calculated interval.

Example: Confidence Interval Using Birth Weights

Given sample statistics: $n = 15$, $\bar{x} = 30.9$ g, $s = 2.9$ g, and a 95% confidence level.

1. Find $t_{\alpha/2}$ for $n = 15$ and 95% confidence level using the T Table.

2. Calculate $E = t_{\alpha/2} \times \dfrac{2.9}{\sqrt{15}}$.

3. Construct the interval: $30.9 - E < \mu < 30.9 + E$.

Z Distribution vs. T Distribution

Choosing the Appropriate Distribution

• Z Distribution: Used when estimating a population proportion $p$ or when the population standard deviation $\sigma$ is known and the sample size is large ($n \geq 30$).

• T Distribution: Used when estimating a population mean $\mu$ with unknown $\sigma$ and small sample size ($n < 30$). The T distribution is wider and accounts for extra uncertainty due to estimating $\sigma$ from the sample.

Degrees of Freedom: For the T distribution, degrees of freedom are $df = n - 1$.

Finding Critical Values $t_{\alpha/2}$

Critical values depend on the confidence level and sample size. Use the T Table to find $t_{\alpha/2}$ for:

• Sample Size $n = 12$, Confidence Level = 95%

• Sample Size $n = 15$, Confidence Level = 90%

• Sample Size $n = 23$, Confidence Level = 98%

• Sample Size $n = 100$, Confidence Level = 80%

Summary Table: Confidence Interval Construction for a Mean

Step	Description
1	Identify sample mean ($\bar{x}$), sample standard deviation ($s$), sample size ($n$), and confidence level
2	Find critical value $t_{\alpha/2}$ from T Table
3	Calculate margin of error $E = t_{\alpha/2} \times \dfrac{s}{\sqrt{n}}$
4	Construct confidence interval: $\bar{x} - E < \mu < \bar{x} + E$
5	Interpret: State confidence that $\mu$ is within the interval

Key Formulas

• Margin of Error for Mean: $E = t_{\alpha/2} \times \dfrac{s}{\sqrt{n}}$

• Confidence Interval for Mean: $\bar{x} - E < \mu < \bar{x} + E$

Applications

Confidence intervals are widely used in research and industry to estimate population parameters, such as average weights, test scores, or proportions, based on sample data. The choice of confidence level affects the width of the interval and the degree of certainty in the estimate.

Additional info: The notes also reference constructing confidence intervals for proportions, but the main focus here is on means using the T distribution. The process is similar for proportions, but uses the Z distribution and different formulas.