Copyright © 2014 Pearson Education. All rights reserved Estimating Population Means LEARNING GOAL Learn to estimate population means and compute the associated margins of error and confidence intervals.
Copyright © 2014 Pearson Education. All rights reserved Slide Estimating a Population Mean: The Basics When we have only a single sample, the sample mean is the best estimate of the population mean, μ. However, we do not expect the sample mean to be equal to the population mean, because there is likely to be some sampling error. Therefore, in order to make an inference about the population mean, we need some way to describe how well we expect it to be represented by the sample mean. The most common method for doing this is by way of confidence intervals.
Copyright © 2014 Pearson Education. All rights reserved Slide A precise calculation shows that if the distribution of sample means is normal with a mean of μ, then 95% of all sample means lie within 1.96 standard deviations of the population mean; for our purposes in this book, we will approximate this as 2 standard deviations. A confidence interval is a range of values likely to contain the true value of the population mean.
Copyright © 2014 Pearson Education. All rights reserved Slide % Confidence Interval for a Population Mean The margin of error for the 95% confidence interval is where s is the standard deviation of the sample. We find the 95% confidence interval by adding and subtracting the margin of error from the sample mean. That is, the 95% confidence interval ranges from (x – margin of error) to (x + margin of error) We can write this confidence interval more formally as x – E < μ < x + E or more briefly as x ± E margin of error = E ≈ 2s n2s n ¯ x ¯ x ¯ x ¯ x ¯ x
Copyright © 2014 Pearson Education. All rights reserved Slide
Copyright © 2014 Pearson Education. All rights reserved Slide Compute the margin of error and find the 95% confidence interval for the protein intake sample of n = 267 men, which has a sample mean of x = 77.0 grams and a sample standard deviation of s = 58.6 grams. EXAMPLE 1 Computing the Margin of Error ¯ x
Copyright © 2014 Pearson Education. All rights reserved Slide Interpreting the Confidence Interval
Copyright © 2014 Pearson Education. All rights reserved Slide EXAMPLE 2 Constructing a Confidence Interval A study finds that the average time spent by eighth-graders watching television is 6.7 hours per week, with a margin of error of 0.4 hour (for 95% confidence). Construct and interpret the 95% confidence interval.
Copyright © 2014 Pearson Education. All rights reserved Slide Choosing Sample Size Choosing the Correct Sample Size In order to estimate the population mean with a specified margin of error of at most E, the size of the sample should be at least where σ is the population standard deviation (often estimated by the sample standard deviation s). E E Solve the margin of error formula for n. E ≈ 2s / n
Copyright © 2014 Pearson Education. All rights reserved Slide Solution: EXAMPLE 3 Constructing a Confidence Interval You want to study housing costs in the country by sampling recent house sales in various (representative) regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size (at a minimum) should be used to ensure that the sample mean is within a. $500 of the true population mean? b. $100 of the true population mean?