4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean

Interpret a confidence interval in context for a population mean or population mean difference.

• Because the confidence interval for a population mean or population mean difference is calculated based on a sample from a population, the computed interval may or may not contain the value of the population mean or population mean difference.
• The interpretation of the confidence level is as follows: In repeated random sampling with the same sample size from the same population, approximately C% of confidence intervals created will capture the population mean or population mean difference, where C represents the numerical value of the confidence level used.
• When interpreting a C% confidence interval for a population mean or population mean difference, we say we are C% confident the interval ()ab, contains the value of the population mean or population mean difference, where a represents the lower limit and b represents the upper limit. An interpretation of a confidence interval for a population mean or population mean difference includes a reference to the parameter.

Justify a claim based on a confidence interval for a population mean or population mean difference.

• A confidence interval for a population mean or population mean difference provides an interval of values that may serve as convincing evidence to support a particular claim about the population mean or population mean difference.

Identify the relationships among sample size, confidence interval width, confidence level, and margin of error for a population mean or population mean difference.

• For a given sample, increasing the confidence level will result in the following:
  • i. The critical value will increase.
  • ii. The margin of error will increase.
  • iii. The width of the confidence interval will increase.
• Increasing the sample size decreases the standard error. Thus, when all other things remain the same, the width of a confidence interval for a population mean or population mean difference tends to decrease as the sample size increases. For a confidence interval for a population mean or population mean difference with a given confidence level, the width of the interval is approximately proportional to 1 n . 127 Inference for Quantitative Data: Means UNIT 4