MACT 3224/317: Assignment N 5 Yahia El Horbaty, Khouzeima Moutanabbir Abd-Elnasser Saad and Noha Youssef Fall 2016 Return this page with your signature Name: Student ID: Section number: I certify that this is my own work, and that I have not copied the work of another student. Date: Signature: Please follow the instructions below Write your name and section number at the spaces provided. Give clear answers and explanations justifying your method wherever possible. Due Thursday, December 15, 2016 at 2:00 pm. Submit your work to our teaching assistant. Homework cannot be submitted as an e-mail attachment. Late submissions are not allowed. The Department of Mathematics and Actuarial Science, The American University in Cairo. 1 Question 1: An investigator computes a 95% confidence interval for a population mean on the basis of a sample of size 70. If she wishes to compute a 95% confidence interval that is half as wide, how large a sample does she need? Question 2: A random sample of the number of barrels (in millions) of oil produced per day by world oil producing countries is listed here: 3.56 1.90 7.83 2.83 1.91 5.88 2.91 6.08 (a) Find the sample mean and the sample standard deviation. (b) Estimate the mean oil production per day with 95% confidence. Question 3: An electrical engineer wants to compare the mean melting points for two dierent metal alloys used in soldering. He collects a sample of size n1 = 36 for the melting point of alloy 1 and finds a sample average 185 o C. He collects a sample of size n2 = 64 for melting point of alloy 2 and finds a sample average of 183 o C. Assume that the population standard deviation for the melting point of the first alloy is 1 = 3 o C and the population standard deviation for the melting point of the second alloy is o 2 = 4 C. (a) Compute a 95% confidence interval for the dierence of the means 1 (b) Using a hypothesis test at level 0.05, is there evidence to claim that 1 2 . 2 = 1o C? Question 4: A sample of 64 resistors are taken from a production line and are found to have a sample mean of 998 Ohms. The population distribution for the resistances is unknown but the population standard deviation is known to be 10 Ohms. (a) Find a 98% confidence interval for the population mean. (b) Answer the following two questions by writing either 'narrower' or 'wider' in the blank space and justify your answer 1. If the sample size were increased to n = 1000 then the confidence interval would be the confidence interval computed from a sample with n = 64. 2. A 99% confidence interval is than a 98% confidence interval. than Question 5: A manufacturer claims that the average life of their electrical light bulbs is 2000 hours. A random sample of 64 bulbs is tested and the life,x, in hours is recorded. The results obtained are as follows Pn - i=1 xi = 127808 Pn - x)2 = 9694.6 i=1 (xi (a) Is there sufficient evidence, at the 1%, that supports the manufacturer's claims. (b) Is there sufficient evidence, at the 1%, that the manufacturer overestimates the length of life of the light bulb. Question 6: In communications let p denote the probability that a bit sent by the transmitter will be received erroneously on the other end of the channel. A telecom company has determined that for a communication channel an error rate of more than 0.05 is unacceptable. To perform the test, a predetermined 32 bit long message is sent (this is a sample with n = 32) and the number of erroneous bits are counted at the receiving end. The number of erroneous bits is 2. 2 (a) Find a 99% confidence interval for the true probability p. (b) Using a hypothesis test at level 0.01, can we conclude that the data present sufficient evidence to suggest that the rate p is unacceptable? Question 7: A questionnaire of spending habits was given to a random sample of college students. Each student was asked to record and report the amount of money they spent on textbooks in a semester. The sample of 130 students resulted in an average of $422 with standard deviation of $57. 1. Give a 90% confidence interval for the mean amount of money spent by college students on textbooks. 2. How many students should you sample if you want a margin of error of $5 for a 90% confidence interval? 3