Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 8.1
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objectives 1. Construct and interpret confidence intervals for a population mean when the population standard deviation is known 2. Find critical values for confidence intervals 3. Describe the relationship between the confidence level and the margin of error 4. Find the sample size necessary to obtain a confidence interval of a given width 5. Distinguish between confidence and probability
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. O BJECTIVE 1 Construct and interpret confidence intervals for a population mean when the population standard deviation is known
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Point Estimate and Margin of Error
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 95% Confidence Interval We are 95% confident that the population mean is between and
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Confidence Level 95% of confidence intervals would cover the true value of the mean
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Terminology A point estimate is a single number that is used to estimate the value of an unknown parameter. A confidence interval is an interval that is used to estimate the value of a parameter. The confidence level is a percentage between 0% and 100% that measures the success rate of the method used to construct the confidence interval. The margin of error is computed by multiplying the critical value by the standard error.
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. O BJECTIVE 2 Find critical values for confidence intervals
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Finding the Critical Value Although 95% is the most commonly used confidence level, sometimes we will want to construct a confidence interval with a different level. We can construct a confidence interval with any confidence level between 0% and 100% by finding the appropriate critical value for that level. The critical values for several common confidence levels are given in the bottom row of Table A.3.
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example – Confidence Interval
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example – Critical Value
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Assumptions
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example – Construct Confidence Interval
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Confidence Intervals on the TI-84 PLUS
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example – Construct Confidence Interval (TI-84 PLUS)
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. O BJECTIVE 3 Describe the relationship between the confidence level and the margin of error
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Confidence Versus Margin of Error If we want to be more confident that our interval contains the true value, we must increase the critical value, which increases the margin of error. There is a trade-off. We would rather have a higher level of confidence than a lower level, but we would also rather have a smaller margin of error than a larger one. 70% Confidence Level Although the Margin of Error is smaller, they cover the population mean only 70% of the time. 95% Confidence Level This represents a good compromise between reliability and margin of error for many purposes. 99.7% Confidence Level They almost always succeed in covering the population mean, but their margin of error is large.
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Measuring the Success Rate of the Method 95% of confidence intervals would cover the true value of the mean
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. O BJECTIVE 4 Find the sample size necessary to obtain a confidence interval of a given width
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sample Size
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. O BJECTIVE 5 Distinguish between confidence and probability
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Interpreting a Confidence Level
Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. You Should Know… How to construct and interpret confidence intervals for a population mean when the population standard deviation is known How to find critical values for confidence intervals How to describe the relationship between the confidence level and the margin of error How to find the sample size necessary to obtain a confidence interval of a given width The difference between confidence and probability