Chapter Confidence Intervals 1 of 31 6  2012 Pearson Education, Inc. All rights reserved.

Chapter Outline 6.1 Confidence Intervals for the Mean (Large Samples) 6.2 Confidence Intervals for the Mean (Small Samples) 6.3 Confidence Intervals for Population Proportions 6.4 Confidence Intervals for Variance and Standard Deviation © 2012 Pearson Education, Inc. All rights reserved. 2 of 31

Section 6.1 Confidence Intervals for the Mean (Large Samples) © 2012 Pearson Education, Inc. All rights reserved. 3 of 31

Section 6.1 Objectives Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine the minimum sample size required when estimating μ © 2012 Pearson Education, Inc. All rights reserved. 4 of 31

Point Estimate for Population μ Point Estimate A single value estimate for a population parameter Most unbiased point estimate of the population mean μ is the sample mean Estimate Population Parameter… with Sample Statistic Mean: μ © 2012 Pearson Education, Inc. All rights reserved. 5 of 31

Example: Point Estimate for Population μ A social networking website allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 40 users of the website. Find a point estimate of the population mean, . (Source: Facebook) © 2012 Pearson Education, Inc. All rights reserved. 6 of 31

Solution: Point Estimate for Population μ The sample mean of the data is Your point estimate for the mean number of friends for all users of the website is friends. © 2012 Pearson Education, Inc. All rights reserved. 7 of 31

Interval Estimate Interval estimate An interval, or range of values, used to estimate a population parameter. Point estimate 12.4 How confident do we want to be that the interval estimate contains the population mean μ? ( ) Interval estimate © 2012 Pearson Education, Inc. All rights reserved. 8 of 31

Level of Confidence Level of confidence c The probability that the interval estimate contains the population parameter. z z = 0-zc-zc zczc Critical values ½(1 – c) c is the area under the standard normal curve between the critical values. The remaining area in the tails is 1 – c. c Use the Standard Normal Table to find the corresponding z-scores. © 2012 Pearson Education, Inc. All rights reserved. 9 of 31

zczc Level of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean μ. z z = 0zczc The corresponding z-scores are c = 0.90 ½(1 – c) = z c = z c = © 2012 Pearson Education, Inc. All rights reserved. 10 of 31

Sampling Error Sampling error The difference between the point estimate and the actual population parameter value. For μ: – the sampling error is the difference – μ –μ is generally unknown – varies from sample to sample © 2012 Pearson Education, Inc. All rights reserved. 11 of 31

Margin of Error Margin of error The greatest possible distance between the point estimate and the value of the parameter it is estimating for a given level of confidence, c. Denoted by E. Sometimes called the maximum error of estimate or error tolerance. When n  30, the sample standard deviation, s, can be used for . © 2012 Pearson Education, Inc. All rights reserved. 12 of 31

Example: Finding the Margin of Error Use the social networking website data and a 95% confidence level to find the margin of error for the mean number of friends for all users of the website. Assume the sample standard deviation is about © 2012 Pearson Education, Inc. All rights reserved. 13 of 31

zczc Solution: Finding the Margin of Error First find the critical values z zczc z = z c = % of the area under the standard normal curve falls within 1.96 standard deviations of the mean. (You can approximate the distribution of the sample means with a normal curve by the Central Limit Theorem, because n = 40 ≥ 30.) z c = 1.96 © 2012 Pearson Education, Inc. All rights reserved. 14 of 31

Solution: Finding the Margin of Error You don’t know σ, but since n ≥ 30, you can use s in place of σ. You are 95% confident that the margin of error for the population mean is about 16.4 friends. © 2012 Pearson Education, Inc. All rights reserved. 15 of 31

Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ The probability that the confidence interval contains μ is c. © 2012 Pearson Education, Inc. All rights reserved. 16 of 31

Constructing Confidence Intervals for μ Finding a Confidence Interval for a Population Mean (n  30 or σ known with a normally distributed population) In WordsIn Symbols 1.Find the sample statistics n and. 2.Specify , if known. Otherwise, if n  30, find the sample standard deviation s and use it as an estimate for . © 2012 Pearson Education, Inc. All rights reserved. 17 of 31

Constructing Confidence Intervals for μ 3.Find the critical value z c that corresponds to the given level of confidence. 4.Find the margin of error E. 5.Find the left and right endpoints and form the confidence interval. Use the Standard Normal Table. Left endpoint: Right endpoint: Interval: In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 18 of 31

Example: Constructing a Confidence Interval Construct a 95% confidence interval for the mean number of friends for all users of the website. Solution:Recall and E= < μ < Left Endpoint:Right Endpoint: © 2012 Pearson Education, Inc. All rights reserved. 19 of 31

Solution: Constructing a Confidence Interval < μ < With 95% confidence, you can say that the population mean number of friends is between and © 2012 Pearson Education, Inc. All rights reserved. 20 of 31

Example: Constructing a Confidence Interval σ Known A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, the standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age. © 2012 Pearson Education, Inc. All rights reserved. 21 of 31

zczc Solution: Constructing a Confidence Interval σ Known First find the critical values z z = 0zczc c = 0.90 ½(1 – c) = z c = z c = © 2012 Pearson Education, Inc. All rights reserved. 22 of 31

Margin of error: Confidence interval: Solution: Constructing a Confidence Interval σ Known Left Endpoint:Right Endpoint: 22.3 < μ < 23.5 © 2012 Pearson Education, Inc. All rights reserved. 23 of 31

Solution: Constructing a Confidence Interval σ Known 22.3 < μ < 23.5 ( ) With 90% confidence, you can say that the mean age of all the students is between 22.3 and 23.5 years. Point estimate © 2012 Pearson Education, Inc. All rights reserved. 24 of 31

Interpreting the Results μ is a fixed number. It is either in the confidence interval or not. Incorrect: “There is a 90% probability that the actual mean is in the interval (22.3, 23.5).” Correct: “If a large number of samples is collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain μ. © 2012 Pearson Education, Inc. All rights reserved. 25 of 31

Interpreting the Results The horizontal segments represent 90% confidence intervals for different samples of the same size. In the long run, 9 of every 10 such intervals will contain μ. μ © 2012 Pearson Education, Inc. All rights reserved. 26 of 31

Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean  is If  is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members. © 2012 Pearson Education, Inc. All rights reserved. 27 of 31

Example: Sample Size You want to estimate the mean number of friends for all users of the website. How many users must be included in the sample if you want to be 95% confident that the sample mean is within seven friends of the population mean? Assume the sample standard deviation is about © 2012 Pearson Education, Inc. All rights reserved. 28 of 31

zczc Solution: Sample Size First find the critical values z c = 1.96 z z = 0zczc z c = z c = 1.96 © 2012 Pearson Education, Inc. All rights reserved. 29 of 31

Solution: Sample Size z c = 1.96   s = 53.0 E = 7 When necessary, round up to obtain a whole number. You should include at least 221 users in your sample. © 2012 Pearson Education, Inc. All rights reserved. 30 of 31

Section 6.1 Summary Found a point estimate and a margin of error Constructed and interpreted confidence intervals for the population mean Determined the minimum sample size required when estimating μ © 2012 Pearson Education, Inc. All rights reserved. 31 of 31