Confidence Intervals for Population Means: Estimation and Calculation, Study Guides, Projects, Research of Statistics

Download Confidence Intervals for Population Means: Estimation and Calculation and more Study Guides, Projects, Research Statistics in PDF only on Docsity! 10. Introduction to Estimation Confidence Interval for a Population Mean • Statistical inference is the process by which we acquire information about populations from samples. • There are two procedures for making inferences: – Estimation – Hypotheses testing Concepts of Estimation • The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. • There are two types of estimators – Point Estimator – Interval estimator Point Estimator A point estimator draws inference about a population by estimating the value of unknown parameter using a single value or a point. An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. An unbiased estimator is said to be consistent if difference between estimator and the parameter grows smaller as sample size grows larger. If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient. Interval Estimator An interval estimator draws inferences about population by estimating the value of an unknown parameter using an interval. 1 Estimating the Population Mean When the Population Standard Deviation is Known • How is an interval estimator produced from a sampling distribution? – To estimate  , a sample of size n is drawn from the population, and its mean X is calculated. – Under certain conditions, X is normally distributed (or approximately normally distributed.), thus n X Z    is standard normally distributed random variables. – Using this fact one can show that              12/2/ n zX n zP and             12/2/ n zX n zXP Interval Estimator of  n zX n zX   2/2/ ,  The probability 1 is called the confidence level. n zX   2/ is called the lower confidence limit. n zX   2/ is called the upper confidence limit. We often represent the interval estimator as n zX   2/ Three commonly used confidence levels 1  2  2 z .90 .10 .05 1.645 .95 .05 .025 1.96 .99 .01 .005 2.575 2 Exercises 1. A physician wanted to estimate the mean length of time  that a patient had to wait to see him after arriving at the office. A random sample of 50 patients showed a mean waiting time of 23.4 minutes and a standard deviation of 7.1 minutes. Find a 95% confidence interval for  . 2. The owner of a small computer company wished to estimate the mean download rate  for the company's update to one of its programs. Forty-five downloads gave a mean rate of 3.1 and a standard deviation of 1.6 kilobits per second. Find a 95% confidence interval for  . 3. Barbara wanted to estimate the mean connection time  to the Internet. She connected 38 times with a mean of 42 and a standard deviation of 5.2 minutess. Find a 90% confidence interval for  . 4. A study was done to estimate the mean annual growth  in a population of Conuspennaceus trees in Hawaii. For those with an initial size of 2.41-2.60 centimeters, a sample of size 33 yielded a mean annual growth of .72 centimeter and a standard deviation of .31 centimeter. Find a 90% confidence interval for the population mean  of annual growth (of those trees with an initial size of 2.41-2.60 centimeters). 5. Sixty pieces of a plastic are randomly selected, and the breaking strength of each piece is recorded in pounds per square inch. Suppose that: X = 26 and s = 1.5 pounds per square inch. Find a 99% confidence interval for the mean breaking strength  . If you were to obtain 200 99% confidence intervals for  , about how many can be expected to contain  ? 6. A city assessor wished to estimate the mean income per household. The previous mean income was $25,300. A random sample of 40 households in the city showed a mean income of $29,400 and a standard deviation of $6325. (a) Find a 95% confidence interval for  , the population mean income per household in the city. (b) Based on your answer in part (a), would the assessor conclude that the mean income had increased over the previous estimate of $25,300? 7. An electrical company tested a new type of oil to be used in its transformers. Thirty- five readings of dielectric strength were obtained. Dielectric strength is the potential (in kilovolts per centimeter of thickness) necessary to cause a disruptive discharge of electricity through an insulator. The results of the test gave: X = 77 kV, s = 8 kV. (a) Find a 95% confidence interval for the mean dielectric strength of the oil. (b) The old transformer oil had a mean dielectric strength of 75 kV. Would you conclude that the new oil has a higher mean dielectric strength on the basis of your answer in part (a)? 5 8. Noise level tests were done on 40 new light rail vehicles (LRVs-the new name for trolley cars). The results of the test gave a sample mean of 65 decibels and a sample standard deviation of 6 decibels. (a) Find a 90% confidence interval for the mean decibel level  for this type of transit vehicle. (b) Based on your answer in part (a), would you conclude that the new LRVs are quieter on the average than older-type trolley cars that had a mean decibel level of 80? 9. An educator wishes to estimate the mean number of hours  that 10-year-old children in a city watch television per day. How large a sample is needed if the educator wants to estimate  to within .5 hour with 90% confidence? Use  = 1.75. 10. How many households in a large town should be randomly sampled to estimate the mean number of dollars spent per household (per week) on food supplies to within $3 with 80% confidence? Assume a standard deviation of $15. 11. Consider a population with unknown mean  and population standard deviation  = 20. (a) How large a sample size is needed to estimate  to within four units with 90% confidence? (b) Suppose that you wanted to estimate  to within four units with 95% confidence. Without calculating, would the sample size required be larger or smaller than that found in part (a)? (c) Suppose that you wanted to estimate  to within two units with 90% confidence. Without calculating, would the sample size required be larger or smaller than that found in part (a)? 12. A production manager noticed that the mean time to complete a job was 160 minutes. The manager made some changes in the production process in an attempt to reduce the mean time to finish the job. A stem-and-leaf plot of a sample of 11 times is as follows: 13 | 9 14 | 25 15 | 01356 16 | 24 17 | 0 Note: 14|5 = 145 minutes The sample mean and standard deviation are 153.36 and 9.47, respectively. Construct a 95% confidence interval for the mean time. 13. A manufacturer claimed that her company's product would not require repair for more than 18 months on the average. A sample of 12 customers who had purchased her product provided the following information on how many months elapsed before repair was needed on their purchases: 6 16.5 17.0 17.5 18.0 18.5 18.5 18.5 19.0 19.0 19.5 20.0 20.5 (a) Construct a stem-and-leaf plot (let 18 | 5 = 18.5). (b) The sample mean and standard deviation times are 18.542 and 1.177, respectively. Construct a 95% confidence interval for  . Does the CI support the belief that the mean repair time is more than 18 months? 14. A psychologist wanted to estimate the mean self-esteem level  of his patients. Fourteen patients were given a test designed to measure self-esteem. The sample mean and standard deviation were 25.3 and 5.3, respectively. Assume the population is approximately normal. (a) Construct a 98% confidence interval for  . (b) Using the confidence interval, could you conclude that  is smaller than the norm of 28.5? 7