A Statistics professor has observed that for several years students score an average of 105 points out of 150 on the semester exam. A salesman suggests that he try a statistics software package that gets students more involved with computers, predicting that it will increase students’ scores. The software is expensive, and the salesman offers to let the professor use it for a semester to see if the scores on the final exam increase significantly. The professor will have to pay for the software only if he chooses to continue using it. ,a) Is this a one- tailed or two- tailed test? Explain. ,b) Write the null and alternative hypotheses. ,c) In this context, explain what would happen if the professor makes a Type I error. ,d) In this context, explain what would happen if the professor makes a Type II error.,e) What is meant by the power of this test?
a. Use a computer to draw 200 random samples, each of size 10, from the normal probability distribution with mean 100 and standard deviation 20. ,b. Find the mean for each sample. ,c. Construct a frequency histogram of the 200 sample means. ,d. Describe the sampling distribution shown in the histogram in part c.
A specialist in the Human Resources department of a national hotel chain is looking for ways to improve retention among hotel staff. The problem is particularly acute among those who maintain rooms, work in the hotel restaurant, and greet guests. Within this chain, among those who greet and register guests at the front desk, the annual percentage who quit is 36% (see the accompanying bar chart for more information).17 Among the employees who work the front desk, more than half are expected to quit during the next year. The specialist in HR has estimated that the turnover rate costs $20,000 per quitter, with the cost attributed to factors such as,? The time a supervisor spends to orient and train a new employee,? The effort to recruit and interview replacement workers,? The loss of efficiencies with a new employee rather than one who is more experienced and takes less time to complete tasks,? Administrative time both to add the new employee to the payroll and to remove the prior employee,To increase retention by lowering the quit rate, the specialist has formulated a benefits program targeted at employees who staff the front desk. The cost of offering these benefits averages $2,000 per employee. The chain operates 225 hotels, each with 16 front-desk employees. As a test, the specialist has proposed extending improved benefits to 320 employees who work the front desk in 20 hotels.,Motivation,(a) Why would it be important to test the effect of the employee benefits program before offering it to all front-desk employees at the hotel chain?,(b) If the benefits program is to be tested, how would you recommend choosing the hotels? How long will the test take to run? (There is no best answer to this question; do your best to articulate the relevant issues.),Method,(c) An analyst proposed testing the null hypothesis H0 : p ? 0.36, where p is the annual quit rate for employees who work the main desk if the new program is implemented. Explain why this is not the right null hypothesis.,(d) Another analyst proposed the null hypothesis H0: p ? 0.36. While better than the choice in part,(c), what key issue does this choice of H0 ignore? What is needed in order to improve this null hypothesis?,Mechanics,(e) If the chosen null hypothesis is H0: p ? 0.30, what percentage of these 320 must stay on (not quit) in order to reject H0 if ? = 0.05?,(f) Assume the chosen null hypothesis is H0: p ? 0.30. Suppose that the actual quit rate among employees who receive these new benefits is 25%. What is the chance that the test of H0 will correctly reject H0?,Message,(g) Do you think that the owners of this hotel chain should run the test of the proposed benefits plan? Explain your conclusion without using technical language.
The purpose of this exercise is to learn how to calculate stock returns for portfolio models using actual stock price data. First, it is necessary to obtain stock price data. One source (of many) is Yahoo! Go to the link http://finance.yahoo.com and type in a ticker symbol such as AAPL (for Apple Computer). Then, on the left-hand side of the page, select Historical Data. ,These data are easily downloaded to a spreadsheet by clicking on the link “Download to Spreadsheet” at the bottom of the page. For Apple Computer (AAPL), Advanced Micro Devices (AMD), and Oracle Corporation (ORCL), download the monthly price data for January 1997 through January 2006. These data contain closing prices that are adjusted for stock dividends and splits.,You now have stock prices for 10 years, and the objective is to calculate the annual returns for each stock for the years 1997 through 2005. Returns are often calculated using continuous compounding. If the stock prices are adjusted for splits and stock dividends, then the price of stock i in period t + 1, Pi,t +1, is given by,pi,t+1 = pt erit, Where pi, t is the price of stock i in period t and rit is the return on stock i in period t. This calculation assumes no cash dividends were paid, which is true of Apple Computer, Advanced Micro Devices, and Oracle Corporation. Solving the equation pi,t+1 = pt erit for the return on stock i in period t gives ,rit = ln (pi,t+1/pt),For example, the Apple Computer adjusted closing price in January 2005 was 38.45. The closing price in January 2006 was 75.51. Thus, the continuously compounded return for Apple Computer from January 2005 to January 2006 is,ln(75.51/38.45) = 0.6749064,We use this calculation as our estimate of the annual return for Apple Computer for the year 2005. ,Take the closing stock prices that you have downloaded and calculate the annual returns for 1997 through 2005 for AAPL, AMD, and ORCL using rit = ln (pi,t+1/pt). If you calculate the returns properly, your results should appear as in Figure.,FIGURE ,YEARLY RETURNS FOR AAPL, AMD, ANDORCL
The profit function for two products is,Profit = -3×12 + 42×1 – 3×22 + 48×2 + 700,Where x1 represents units of production of product 1 and x2 represents units of production of product 2. Producing one unit of product 1 requires 4 labor-hours and producing one unit of product 2 requires 6 labor-hours. Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function. However, it is possible to schedule overtime at a premium of $5 per hour.,a. Formulate an optimization problem that can be used to find the optimal production quantity of products 1 and the optimal number of overtime hours to schedule.,b. Solve the optimization model you formulated in part (a). How much should be produced and how many overtime hours should be scheduled?
A smart meter tracks household electricity use and the time of use. With smart meters, consumers have a price incentive to use electricity at off-peak hours, when rates are cheaper. After having smart meters described to them, a survey asked 2,400 Canadians about their interest level in having a smart meter installed. At the 5% level of significance, is there evidence that fewer than one-quarter of all Canadians are extremely interested or very interested in having smart meters installed in their homes?
A sample of 300 orders for take-out food at a local pizzeria found that the average cost of an order was $23 with s = $15.,(a) Find the margin of error for the average cost of an order.,(b) Interpret for management the margin of error.,(c) If we need to be 99% confident, does the confidence interval become wider or narrower?,(d) Find the 99% confidence interval for the average cost of an order.
A sample of 150 calls to a customer help line during one week found that callers were kept waiting on average for 16 minutes with s = 8.,(a) Find the margin of error for this result if we use a 95% confidence interval for the length of time all customers during this period are kept waiting.,(b) Interpret for management the margin of error.,(c) If we only need to be 90% confident, does the confidence interval become wider or narrower?,(d) Find the 90% confidence interval.
A sample of 144 values is randomly selected from a population with mean, m, equal to 45 and standard deviation, s, equal to 18. ,a. Determine the interval (smallest value to largest value) within which you would expect a sample mean to lie. ,b. What is the amount of deviation from the mean for a sample mean of 45.3? ,c. What is the maximum deviation you have allowed for in your answer to part a? ,d. How is this maximum deviation related to the standard error of the mean?