HAWKES LEARNING SYSTEMS math courseware specialists Copyright 2010

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 10 Estimating Means and Proportions

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Sections 10. 1 -10. 3 Introduction to Estimating Means Objectives: • Determine the best point estimates for population parameters. • Choose which method would be most appropriate when calculating the margin of error for the population mean.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 1 What is an Estimator? Definitions: • Estimator– a strategy or rule that is used to estimate a population parameter. • Estimate– the result of applying an estimator to a specific set of data. • Point Estimate – a single number estimate of a population parameter. The best point estimate of a population mean is the sample mean. • Interval Estimate – a range of possible values for the population parameter.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 3 Point Estimation of the Population Mean Definitions: • Unbiased Estimator – an estimator whose expected value is equal to the parameter that is being estimated. • Mean Squared Error (MSE) – an estimator’s average squared distance from the true parameter. • The Mean Squared Error (MSE) for the sample mean is given by: Among unbiased estimators of the population mean, the sample mean has the smallest mean squared error.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Sections 10. 4, 10. 6 Interval Estimation of the Population Mean Objectives: • To learn the meaning of a confidence interval. • To determine the required sample size for a particular confidence level.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 4 Interval Estimation of the Population Mean Definitions: • • Confidence Interval– a particular interval estimate of a population parameter. A Confidence Interval for the Population Mean is given by if either of the following conditions are true. • n > 30 (use s as an approximation for σ) • If σ is known and the population being studied is normal.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 4 Interval Estimation of the Population Mean Definitions: • This expression, creates the “generalized” confidence interval. The is the point estimate, while is the maximum error of estimation with a specific level of confidence, given a sample size n. The term represents the z-value required to obtain an area of 1 -α centered under the standard normal curve. The z-values for obtaining various (1 - α) areas centered under the standard normal curve are given below. Confidence 1−α 0. 80 1. 28 0. 90 1. 645 0. 95 1. 96 0. 99 2. 575

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 4 Interval Estimation of the Population Mean Example: • Construct a 95% confidence interval for the population mean if the standard deviation of the population is 900. Use the following sample data: Solution: 95% Confidence Interval We are 95% confident that the point estimate of μ, has an error of estimation no larger than 176. 4. Being able to assess the error of an estimate is one of the most useful applications of statistical methods.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 6 Precision and Sample Size: • • • The more accurate an estimate, the greater its potential value in decision making. The only way to perfectly determine an unknown population parameter is to perform a census. This is usually impractical because of cost and/or time considerations. The width of the confidence interval defines the precision; the smaller the interval, the greater the precision. The three components which affect the width of the confidence interval for the population mean are: Represents the distance the confidence interval boundary is from the estimated mean in standard deviation units. The distance is related to the specified level of confidence. Represents the population standard deviation. n Represents the sample size.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 6 Precision and Sample Size: • Level of confidence can vary in order to reduce/expand the confidence interval width, but doing so does not increase the information it only presents it differently. • σ, the population standard deviation, is a constant and does not change. • n, the sample size, can vary, and the larger the sample the smaller the width of the resulting confidence interval for some given level of confidence. • How large should the sample size be? The sample size should be selected relative to the size of the maximum positive or negative error the decision maker is willing to accept, given by:

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 6 Precision and Sample Size Example: Suppose that a quality control manager wishes to measure the average amount of cleaning fluid the company puts in their 24 ounce bottles. From previous samples, they believe their standard deviation is. 4 ounces. How large must the sample be in order to be 95% confident of estimating the mean cleaning fluid in a 24 ounce bottle to within. 08 ounces? Solution: • Using we get from which we can get • We need to round 96. 04 up to the next integer, 97, in order to assure the desired level of confidence. With a sample size of 97 we are 95% confident of estimating the mean cleaning fluid in a 24 ounce bottle to within. 08 ounces.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 6 Precision and Sample Size for σ Unknown : • In the previous slides σ was assumed to be known. This assumption is usually unreasonable. • The most obvious method for obtaining an estimate of σ is to take a small sample and use the sample standard deviation as an estimate of the population standard deviation. Replacing σ with s in the sample size formula will provide an initial estimate of the required sample size.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown Objectives: • To determine t if given a probability. • To determine the value of • To construct a confidence interval for a given situation.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown Constructing a Confidence Interval with σ Unknown: • If σ is not known and n 30, the confidence interval must be changed slightly. As long as the population is normally distributed, the distribution of the quantity where s is the standard deviation of the sample, has a Student’s t-distribution.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown The t-Distribution: • The t-distribution is very much like the normal distribution. It is a symmetrical, bell-shaped distribution with slightly thicker tails than a normal distribution. The t-distribution approaches the normal distribution as the degrees of freedom become larger.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown Degrees of Freedom: • The t-distribution has one parameter, degrees of freedom. The degrees of freedom for any t-distribution is computed in the following manner. Confidence Interval for the Population Mean: • If σ is unknown and the sample is drawn from a normal population, a (1 -α) confidence interval for the population mean is given by where is the critical value for a t-distribution with n− 1 degrees of freedom which captures an area of in the right tail of the distribution.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown Example: • • Given the following data drawn from a normal population with unknown mean and variance, construct a 95% confidence interval for the population mean. Seven data values have been selected randomly from the population. 25, 19, 37, 29, 40, 28, 31 Solution: • The sample mean and standard deviation are , respectively. The degrees of freedom are d. f. = n − 1 = 7 − 1 = 6. and

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 5 Interval Estimation of the Population Mean for a Normal Population with σ Unknown Solution: • The t-value corresponding to 6 degrees of freedom and 95% confidence is given in Table D of Appendix A as 2. 447. The corresponding confidence interval is Thus, we are 95 percent confident that the interval will contain the population mean. • An alternate interpretation would be that we are 95% confident that the point estimate (29. 86) has a maximum error of estimation of 6. 55. =

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Sections 10. 7 -10. 9 Estimation (Proportions) Objectives: • To calculate the minimum sample size for proportions. • To estimate the standard deviation of proportions.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 7 Estimating Population Attributes: • • An attribute is a characteristic that members of a population either possess or do not possess. Attributes are almost always measured as the proportion of the population that possess the characteristic. Examples include: the percentage of television viewers who are watching a particular program, the fraction of teachers who believe group learning is a beneficial instructional method, and the percentage defective in a lot of goods.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 7 Estimating Population Attributes: • Estimating the proportion of the population that possess an attribute is straightforward. A random sample is selected and the sample proportion is computed as follows: • The symbol above the p indicates an estimate of the quantity specified.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 7 Estimating Population Attributes Example: • Estimate the fraction of defective transistors in a lot containing 10, 000 transistors. Suppose a sample size of 400 is drawn from the lot, and 3 transistors were found to be defective. Solution: • Let Then The question though, is “How good is the estimate of the fraction of defective transistors? ” The answer to this question arises in the discussion of interval estimation for proportions.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 8 Interval Estimation of a Population Attribute: • • • The concept of confidence intervals can also be applied to estimating proportions. In order to develop the confidence interval for a population proportion, the sampling distribution of the point estimate must be developed. The sample proportion, , is distributed normally with mean, p, and variance, If the sample size is sufficiently large, and the (1 -α) confidence interval for the population proportion is given by the expression • Where is the distance from the point estimate to either end of the interval in standard deviation units, and is the standard deviation of

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 8 Interval Estimation of a Population Attribute Example: • Suppose a sample of 410 randomly selected radio listeners revealed that 48 listened to WXQI. This is a point estimate of the proportion that listen to WXQI. Solution: • To obtain an interval estimate, the amount of confidence to be placed in the interval must be specified. Suppose we desire 95% confidence.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 8 Interval Estimation of a Population Attribute Solution: • • Note on the previous slide that the sample proportion is used in place of p in the computation of For any realistic problem, this will always be the case. Fortunately, unless and are far apart, the value of will not be greatly affected. Computing the confidence interval results in We are 95% confident that the point estimate, 0. 117 has a maximum error of estimation of 0. 03116. A maximum error of only 0. 03116 with 95% confidence suggests a rather high level of accuracy in the estimation of the proportion.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 9 Precision and Sample Size of Population Attributes: • Just as for the population mean, a specific level of accuracy in estimating a population proportion is desirable. In order to estimate extremely small quantities, highly precise estimates are necessary. The technique for deriving the sample size parallels the discussion of the sample mean. Setting one-half of the entire width of the confidence interval equal to the maximum allowable error yields Solving for n yields

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 9 Precision and Sample Size of Population Attributes: • Usually the population proportion is unknown and is estimated from a previous study. In this case the sample size is given by where is the estimate of the population proportion obtained from the previous study. • If an estimate of the population proportion is not available, then the population proportion is set equal to. 5. The value. 5 maximizes the quantity and thus provides the most conservative estimate of the sample size possible. Remember to always round the sample size to the next largest integer to assure the desired level of accuracy.

HAWKES LEARNING SYSTEMS math courseware specialists Estimating Means and Proportions Section 10. 9 Precision and Sample Size of Population Attributes Example: • How large of a sample would be required to estimate the proportion of buyers on a mailing list with an accuracy of 0. 002, a 95% degree of confidence, and a previous proportion of 0. 008. Solution: • Using the sample size formula yields Thus, to be 95% confident that the proportion is estimated with an error of at most 0. 002 requires a sample size of 7, 622.