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STATISTICS 101: Section L - Homework 6 Due Friday, April 2, 2004 • Homework is due by 4:00 pm on the due date. You can always turn in your homework in class. • You may talk with others about the homework problems but please write your solutions up independently. • Please answer homework questions in complete sentences. Make sure to staple the pages of your assignment together. • You will have an opportunity to get help on homework during lab. Reading: Mar. 22 - Mar. 24 Section 5.2 Mar. 26 - Apr. 2 Chapter 6 Assignment: 1. During a day, a diet conscious individual consumes food that has a total caloric content, X, that is normally distributed with mean, µ = 1850 calories and standard deviation, σ = 120 calories. (a) For a single day, chosen at random, what is the probability that the food consumed by this individual will contain more than 2000 calories? (b) For a random sample of 25 days, describe the sampling distribution of X, the sample mean caloric content of food consumed. Does this result depend on the Central Limit Theorem? Explain briefly. (c) What is the chance that the sample mean caloric content of food consumed for a random sample of 25 days will be less than 1775 calories? 2. Large packages of potato chips are supposed to contain 13 ounces of product. When the filling process is working correctly the actual weight of contents is approximately normally distributed with mean µ=13.6 ounces and standard deviation σ=0.5 ounces. The mean is set high so that the under-filling will not occur frequently. (a) Find the probability that a randomly selected package contains less than 13 ounces. (b) As part of a quality control effort, the company takes a random sample of 4 packages each day and weighs the contents of each. Find the probability that the sample mean weight for the 4 packages is less than 13 ounces. Explain why the answer here is smaller than in part (a). (c) The company will not ship the days production if the sample mean weight of the 4 packages is too low. What should the cut off value be so that the chance of shipping the days production is 0.99? (d) How does the cut off value change in part (c) if the company wishes to base the decision to ship on a random sample of 16 packages? 1 3. Read pages 414-429 and do exercises 6.6. 4. In order to comply with federal mandates school districts must insure that “No child is left behind.” A school district is looking into using a standardized test to evaluate the performance of fifth graders on basic skills (Math, Science and Language Arts). The standardized test is known to produce scores that are normally distributed with a standard deviation σ = 20 points. (a) In a pilot study, n=16 randomly selected fifth graders take the standardized test. The sample average score is 105.5 points. Construct a 95% confidence interval for the mean score of all fifth graders in the district. (b) A school board member sees the results and wants to be 99% confident. Construct a 99% confidence interval for the mean score of all fifth graders in the district. (c) Which interval is wider, the 95% confidence interval in (a) or the 99% confidence interval in (b)? Why is this the case. (d) The school board member suggests expanding the study to include n=100 randomly selected fifth graders. What is the margin of error for a 99% confidence interval based on n=100? How does this margin of error compare to the margins of error in (a) and (b). (e) The school board member would like to be 99% confident and have a margin of error of 2 pts. How large a random sample should be taken? (Be careful, you can’t chop fifth graders up into pieces.) 5. The number of accidents that occur each day at a large underground mining operation has a distribution that is skewed to the right (many days there are none or only one accident but on some days there are many accidents) with a standard deviation σ = 1.8. A labor group is concerned that new work rules are changing the rate of accidents. Management says that the mean number of accidents per day continues to be 1.2. The labor group obtains a random sample of 25 days of accident reports since the new work rules were instituted. The sample mean number of accidents is X = 2.1. (a) Construct a 95% confidence interval on the population mean number of accidents per day. (b) What does this interval indicate about management’s claim that the mean number of accidents per day is 1.2? (c) Can one conclude that the new work rules caused a change in the mean number of accidents per day? 2