Estimation Theory. (Exercises)

Transcription

1 Chapter II Estimation Theory (Exercises) The Annual salary for public school teachers in the US is $ The mean salary for a sample of 750 school teachers equals Identify the population, the population mean, and the sample mean. Identify the parameter and the point estimate of the parameter Ninety-eight percent of homes in a certain city have a TV. A survey of 2000 homes finds that 1925 have a TV. Identify the population, the population proportion, and the sample proportion. Identify the parameter and point estimate of the parameter When a multiple of the standard error of a point estimate is subtracted from the point estimate to obtain a lower limit and added to the point estimate to obtain a upper limit, what term is used to describe the numbers between the lower limit and the upper limit? For a data set obtained from a sample of 20 items and the mean 24.5, it is known that the population is normally distributed with a standard distribution of What is the point estimate of the population? 2. Make a 99% confidence interval for the mean of the population A sample of 1500 homes sold recently in a town gave the mean price of The population standard deviation was Construct a 99% confidence interval for the mean price of homes in this town A bank manager has developed a new system to reduce the time customers spend waiting to be served by tellers during peak business hours. Typical waiting times during peak business hours under the current system are roughly 9-10 minutes. The bank manager hopes that the new system will lower typical waiting times to less than six minutes. A 30-day trial of the new system is conducted. During the trial run, every 150th customer who arrives during peak business hours is selected until a systemic sample of 100 customers is obtained. Each of the sampled customers is observed, and the time spent waiting for teller service is recorded. Moreover, the bank manager feels that this systemic sample is as representative as a random sample of waiting times would be. The mean of the sample of 100 customers waiting times is Assuming a standard deviation of 2.47 minutes for all customers, 1

2 1. Calculate 95% and 99% confidence intervals for the mean of the waiting time of all customers during the peak hours, 2. Using the 95% confidence interval, can the bank manager be 95% confident that the mean is less than 6 minutes? 3. Using the 99% confidence interval, can the bank manager be 99% confident that the mean is less than 6 minutes? 4. Based on your answers to the questions in parts 1 and 2, how convinced are you that the new mean waiting time is less than 6 minutes? Referring to Exercise 2.6., assume that the standard deviation of waiting times of all customers is unknown. However, the corresponding value for the sample of 100 customers is equal to Calculate a 95% confidence interval for the mean of all possible bank customers waiting times using the new system. Are we 95% confident that this mean is less than six minutes? A marketing research analyst collects data for a random sample of 100 customers out of the 4000 who purchased a particular coupon special. The 100 people spent an average of in the store with a standard deviation of Using a 95 percent confidence interval, estimate 1. the mean purchase amount for all 4000 customers 2. the total euro amount of purchases by the 4000 customers A prospective purchaser wishes to estimate the mean Euro amount of sales per customer at a toy store at an airline terminal. Based on data from other similar airports, the standard deviation of such sales amounts is estimates to be about What size of random sample should be collected, as a minimum, if the purchaser wants to estimate the mean sales amount within 1.00 and with 99% percent confidence? Referring to Exercise 2.9., what is the minimum required sample size if the distribution of sales amount is not assumed to be normal and the purchaser wishes to estimate the mean sales amount within 2.00 with 99 percent confidence? Cigarette filters are supposed to have an average length of 15 mm and a standard deviation 0.1 mm. Machinery will jam if the length of a filter exceeds 15.3 mm, and the probability of such a filter increases as the mean length increases; must have an accurate estimate of the mean length of filters. Let us require an error of at most 0.01 mm. and 90% confidence intervals. How large must the sample size be? As a commercial buyer for a private supermarket brand, suppose you take a random sample of 12 No. 303 cans of string beans at a canning plant. The net weight of the drained beans in each can is reported in the following table: 2

3 Net Weight of Beans in 12 No. 303 Cans Ounces per can Number of cans Determine i) the mean net weight of string beans being packed in each can for this sample, ii) the sample standard deviation. 2. Assuming that the net weights per can are normally distributed, estimate the mean weight per can of beans packed using a 95 percent confidence interval Suppose that you wish to estimate the mean sales amount per retail outlet for a particular consumer product during the past year. The number of retail outlets is large. 1. Determine the 95 percent confidence interval given that the sales amounts are assumed to be normally distributed, x _ = 3425, σ = 200, and n = Determine the 95 percent confidence interval given that the population is assumed to be normally distributed, x _ = 3425, s = 200, and n = Determine the 95 percent confidence interval given that the population is not assumed to be normally distributed, x _ = 3425, s = 200, and n = A random sample of 20 acres gave a mean of 41.2 bushels per acre with a standard deviation of 3 bushels. Assuming that the yield of wheat per acre is normally distributed, construct a 90% confidence interval for the population mean Customers of a car insurance company sometimes have to wait a long time to speak to a customer service representative when they call regarding disputed claims. A random sample of 25 such calls yielded a mean waiting time of 22 minutes with a standard deviation of 6 minutes. Construct a 99% confidence interval for the population mean of such waiting times. Assume that such waiting times for the population follow a normal distribution Suppose that we conduct a survey of 19 millionaires to find out what percent of their income the average millionaire donates to charity. We discover that the mean percent is 15 with a standard deviation of 5 percent. Find a 95% confidence interval for the mean percent A company that manufactures exercising machines wanted to know the percentage of apartments in a community that provide on-site health care and exercise facilities. A sample survey of 119 such apartments showed that 75 of these apartments provide such facilities. 3

4 1. What is the point estimate of the percentage of all such apartments that provide such facilities? 2. Assuming a 95% confidence level, what is the bound on error associated with this point estimate? 3. Find a 95% confidence interval for the percentage of all such apartments that provide such facilities? A major newspaper selected a simple random sample of 1600 readers from their list of subscribers. They asked whether the paper should increase its coverage of local news. Forty percent of the sample wanted more local news. What is the 99% confidence interval for the proportion of readers who would like more coverage of local news? In 2000, a certain survey asked adult US American if they would be willing to pay much higher taxes in order to protect the environment. Of 1167 respondents, 369 indicated a willingness to do so. Find a 99% confidence interval for the population proportion of adult Americans willing to do so when the survey was conducted An electronics company has just installed a new machine that makes a part that is used in computers. The company wants to estimate the proportion of these parts produced by this machine that are defective. The company manager wants this estimate to be within 0.02 of the population proportion for a 95% confidence level. What is the most conservative estimate of the sample size that will limit the maximum error to within 0.02 of the population proportion? Many states in the USA are carefully considering steps that would help them collect taxes on items purchased through the Internet. How many randomly selected sales transactions must be surveyed to determine the percentage that transpired over the Internet? Assume that we want to be 99% confident that the sample percentage is within two percentage points of the true population percentage of all sales transactions? A computer company sells computers and computer parts by mail. The company claims that at least 90% of all orders are mailed within 72 hours after they are received. The quality control department at the company often takes samples to check if this claim is valid. A recently taken sample of 150 orders showed that 129 of them were mailed within 72 hours. Can you conclude that the company s claim is false? Use a 2.5% significance level. 4

5 (Last updated: ) 5