Confidence Intervals for the Mean (σ Known)

Chapter 6: Confidence Intervals

6.1 Confidence Intervals for the Mean (σ Known)

Confidence intervals provide a range of values used to estimate a population parameter, such as the mean. When the population standard deviation (σ) is known, specific methods are used to construct and interpret these intervals.

Point Estimate

• Definition: A point estimate is a single value used to estimate a population parameter. For the population mean (μ), the sample mean (\( \bar{x} \)) is the most unbiased point estimate.

• Example: If a random sample of 40 student-athletes has a mean of 19.6 hours spent on required athletic activities per week, then 19.6 is the point estimate for the population mean.

Interval Estimate

• Definition: An interval estimate is a range of values used to estimate a population parameter. It reflects the uncertainty inherent in using sample data to estimate population values.

• Before calculating the interval, determine the desired level of confidence.

Level of Confidence

• Definition: The level of confidence (c) is the probability that the interval estimate contains the population parameter, assuming the estimation process is repeated many times.

• Common confidence levels are 90%, 95%, and 99%.

Critical Values

• Definition: Critical values are values that separate probable sample statistics from improbable ones, based on the chosen confidence level.

• For a 95% confidence level, the critical z-score is 1.96.

Sampling Error

• Definition: Sampling error is the difference between the point estimate and the actual population parameter. It is generally unknown and varies from sample to sample.

Margin of Error (E)

• Definition: The margin of error (E) is the greatest possible distance between the point estimate and the value of the parameter it is estimating. It is also called the maximum error of estimate or error tolerance.

• Formula (σ known):

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

• Where z_{\frac{\alpha}{2}} is the critical value for the desired confidence level, σ is the population standard deviation, and n is the sample size.

• Example: For a sample mean of 19.6, σ = 1.2, n = 40, and a 95% confidence level (z = 1.96):

$E = 1.96 \cdot \frac{1.2}{\sqrt{40}} \approx 0.4$

Confidence Interval for the Mean (σ Known)

• Formula:

$\left( \bar{x} - E, \; \bar{x} + E \right)$

• Where \( \bar{x} \) is the sample mean and E is the margin of error.

• Example: With \( \bar{x} = 19.6 \) and E = 0.4, the 95% confidence interval is (19.2, 20.0).

Constructing Confidence Intervals Using Technology

• Statistical software (e.g., Minitab, StatCrunch, TI-84 Plus) can be used to construct confidence intervals by entering raw data or summary statistics.

• Example: A 99% confidence interval for the same data might be (19.1, 20.1).

Interpreting Confidence Intervals

• The confidence interval either contains the population mean or it does not; the probability statement refers to the method, not the specific interval.

• Correct interpretation: “If a large number of samples is collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain the population mean.”

• Incorrect interpretation: “There is a 90% probability that the actual mean is in the interval.”

Determining Minimum Sample Size

• To estimate the population mean with a specified margin of error and confidence level, use:

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

• If n is not a whole number, round up to the next whole number.

• If σ is unknown, use the sample standard deviation s from a preliminary sample (n ≥ 30).

• Example: To be 95% confident that the sample mean is within 0.5 hour of the population mean, with σ = 1.2, z = 1.96:

$n = \left( \frac{1.96 \cdot 1.2}{0.5} \right)^2 \approx 22.1$ Round up to 23.

Summary Table: Key Concepts in Confidence Intervals (σ Known)

Concept	Definition	Formula
Point Estimate	Single value estimate for a population parameter	\( \bar{x} \)
Margin of Error (E)	Maximum distance between point estimate and parameter	$E = z_{\frac{\alpha}{2}} \cdot \frac{\sigma}{\sqrt{n}}$
Confidence Interval	Range likely to contain population mean	$\left( \bar{x} - E, \; \bar{x} + E \right)$
Minimum Sample Size (n)	Sample size needed for desired margin of error	$n = \left( \frac{z_{\frac{\alpha}{2}} \cdot \sigma}{E} \right)^2$

Additional info: The image included is the cover of the textbook 'Elementary Statistics: Picturing the World' by Ron Larson, which is directly relevant as it visually identifies the source of the material and reinforces the academic context of the notes.