Estimation and Confidence Intervals

Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean when s is known. Estimate a population mean from a sample mean when s is unknown. Estimate a population proportion from a sample proportion. Estimate the population variance from a sample variance. Estimate the minimum sample size necessary to achieve given statistical goals. 2

Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample. A point estimate is the statistic, computed from sample information, which is used to estimate the population parameter.

Statistical Estimation A confidence interval estimate is a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence. Interval Estimate - a range of values calculated from a sample statistic(s) and standardized statistics, such as the z. Selection of the standardized statistic is determined by the sampling distribution. Selection of critical values of the standardized statistic is determined by the desired level of confidence. 3

Factors Affecting Confidence Interval Estimates The factors that determine the width of a confidence interval are: 1.The sample size, n. 2.The variability in the population, usually σ estimated by s. 3.The desired level of confidence.

Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Upper Confidence Limit Lower Confidence Limit Point Estimate Width of confidence interval

Confidence Intervals How much uncertainty is associated with a point estimate of a population parameter? An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence intervals

Interval Estimates - Interpretation For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population

Estimation Process Random Sample Population I am 95% confident that μ is between 40 & 60. Random Sample Population Mean X = 50 (mean, μ, is unknown) Sample

Confidence Interval to Estimate  when  is Known Point estimate Interval Estimate 4

95% Confidence Interval for  10

95% Confidence Interval for  10

Question: A survey was taken of U.S. companies that do business with firms in India. One of the questions on the survey was: Approximately how many years ha your company been trading with firms in India? A random sample of 44 responses to this question yielded a mean of 10.455 years. Suppose the population standard deviation for this question is 7.7 years. Using this information , construct a 90%condidence interval for the mean number of years that a company has been trading in India for the population of U.S. companies trading with firms in India.

Solution: The analyst is 90% confident that is a census of al U.S. companies trading with firms in India were taken at the time of this survey, the actual population mean number of years a company would have been trading with firms in India would be between 8.545 and 12.365. The point estimate is 10.455 years 13

Question: A study is conducted I a company that employs 800 engineers. A random sample of 50 engineers reveals that the average sample age is 34.3 years. Historically, the population standard deviation of the age of the company’s engineers is approximately 8 years. Construct a 98% confidence interval to estimate the average age of all the engineers in this company.

Solution: The finite correction factor takes into account the fact that the population is only 800 instead of being infinitely large. The sample, n = 50, is a greater proportion of the 800 than it would be of a larger population, and thus the width of the confidence interval is reduced. 14

Confidence Interval to Estimate  when n is Large and  is Known 15

Question: Suppose a U.S. car rental firm wants to estimate the average number of miles traveled per day by each of its car rented in California. A random sample of 110 cars rented in California reveals that the sample mean travel distance per day is 85.5miles, with a sample standard deviation of 19.3miles. Compute a 99% confidence Interval to estimate µ

Solution: 13

Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level z Value 1.645 1.96 2.33 2.575 17

Estimating the Mean of a Normal Population: Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. z distribution is not appropriate for these conditions t distribution is appropriate 18

The t Distribution Developed by British statistician, William Gosset A family of distributions - a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a z t formula 19

Confidence Intervals for  of a Normal Population: Unknown  22

Example A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?

Confidence Interval for the Mean Example using the t-distribution

Student’s t-distribution Table

Question: The owner of a large equipment rental company wants to make a rather quick estimate of the average number of days a piece of ditch-digging equipment is rented out per person per time. The company has record of all accounts world be prohibitive. The owner decides to take a random sample of rental invoices. Fourteen different rentals of ditch-diggers are selected randomly from the files, yielding the following data. She uses these data to construct a 99% confidence interval to estimate the average number of days that a ditch-digger is rented and assumes that the number of days per rental is normally distributed in the population. 3 1 3 2 5 1 2 1 4 2 1 3 1 1

Solution: 23

Solution: 24

Example: There are 250 families in Scandia, Pennsylvania. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75. Develop a 90 percent confidence interval for the population mean. Interpret the confidence interval.

Solution: Given in Problem: N – 250 n – 40 s - $75 Since n/N = 40/250 = 0.16, the finite population correction factor must be used. The population standard deviation is not known therefore use the t-distribution (may use the z-dist since n>30) Use the formula below to compute the confidence interval:

Confidence Interval to Estimate the Population Proportion 25

Problem: A clothing company produces men’s jeans. The jeans are made and sold with either a regular cut or a boot cut. In an effort to estimate the proportion of their men’s jeans market in Oklahoma City that prefers boot-cut jeans, the analyst takes a random sample of 212 jeans sales form the company’s two Oklahoma City retail outlets. Only 34 of the sales were for boot-cut jeans. Construct a 90% confidence interval to estimate the proportion of the population in Oklahoma City who prefer boot-cut jeans.

Solution: 26

Population Variance Variance is an inverse measure of the group’s homogeneity. Variance is an important indicator of total quality in standardized products and services. Managers improve processes to reduce variance. Variance is a measure of financial risk. Variance of rates of return help managers assess financial and capital investment alternatives. Variability is a reality in global markets. Productivity, wages, and costs of living vary between regions and nations. 27

Estimating the Population Variance Population Parameter  Estimator of   formula for Single Variance 28

Confidence Interval for 2 29

90% Confidence Interval for 2 33

Problem: The U.S. Bureau of Labor Statistics publishes data on the hourly compensation costs for production workers in manufacturing for various countries. The latest figures published for Greece show that the average hourly wage for a production worker in manufacturing in $9.63. Suppose the business council of Greece wants to know how consistent this figure is. They randomly select 25 production workers in manufacturing from across the country and determine that the standard deviation of hourly wages for such workers is $1.12. Use this information to develop a 95% confidence interval to estimate the population variance for the hurly wages of production workers in manufacturing in Greece. Assume that the hourly wages for production workers across the country in manufacturing are normally distributed.

Solution: 34

Problem: The Interstate Conference of Employment Security Agencies says the average workweek in the United Sates is down to only 35 hours, largely because of a rise in part-time workers. Suppose this figure was obtained forma a random sample of 20 workers and that the standard deviation of the sample was 4.3 hours. Assume hours worked per week are normally distributed in the population. Use this sample information to develop a 98%confidence interval for the population variance of the number of hours worked per week for a worker.

Solution: n = 20 s = 4.3 s2 = 18.49 98% C.I. df = 20 – 1 = 19 2.99,19 = 7.63273 2.01,19 = 36.1980 < 2 < 9.71 < 2 < 46.03 34

Determining Sample Size when Estimating  z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated  35

Problem: Suppose a researcher wants to estimate the average monthly expenditure on the bread by a family in Chicago. She wants to be 90% confidant of her results. How much error is she willing to tolerate in the results? Suppose she wants the estimate within $1.00 of the actual figure and the standard deviation of average monthly bread purchases is $4.00. what is the sample size estimation for this problem?

Sample Size When Estimating µ: Example 36

Question: Suppose you want to estimate the average age of all Boeing 727 airplanes now in active domestic U.S. service. You want to be 95% confident, and you want your estimate to be within two years of the actual figure. The 727 was first placed in service about 30 years ago, but you believe that no active 727s in the U.S. domestic fleet are more than 25 years old. How large a sample should you take?

Solution: 37

Determining Sample Size when Estimating p z formula Error of Estimation (tolerable error) Estimated Sample Size 38

Question: Hewitt Associates conducted a national survey to determine the extent to which employers are promoting health and fitness among their employees. One of the questions asked was, Does your company offer on-site exercise classes? Suppose it was estimated before the study that no more than 40% of the companies would answer Yes. How large a sample would Hewitt Associates have to take in estimating the population proportion to ensure a 98% confidence in the results and to be within .03 of the true population proportion?

Solution: 39

Determining Sample Size when Estimating p with No Prior Information 0.5 0.4 0.3 0.2 0.1 pq 0.25 0.24 0.21 0.16 0.09 P n 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z = 1.96 E = 0.05 40

Example: Determining n when Estimating p with No Prior Information 41