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Stat 430 Name: Midterm Exam Oct 9, 2008 This is a closed book, closed notes exam. No outside help is permitted. You are allowed to use a calculator. Make sure to write your name legibly on the sheets you hand in. Good luck! Points Page 1 / 25 Page 2 / 25 Page 3 / 25 Page 4 / 25 Total / 100 Stat 430 Name: Midterm Exam Oct 9, 2008 1. Google search response times The following values are the response times of 55 Google searches (in seconds). Please note that the results are sorted. 0.003 0.029 0.049 0.097 0.206 0.003 0.034 0.049 0.117 0.215 0.004 0.041 0.055 0.119 0.216 0.016 0.043 0.063 0.122 0.236 0.017 0.045 0.067 0.129 0.255 0.017 0.045 0.068 0.136 0.283 0.018 0.046 0.078 0.143 0.284 0.018 0.046 0.081 0.153 0.322 0.022 0.046 0.081 0.180 0.358 0.022 0.047 0.083 0.185 0.472 0.023 0.048 0.088 0.197 0.486 The total of all response times is 6.306 sec, the total of all squared response times is 1.4224 sec2 . (a) (6 points) Based on the data, give the following summaries: Minimum: Mean: Median: Mode: Maximum: Range: (b) (5 points) Construct and draw a boxplot - are there outliers? (c) (8 points) Derive a 90% confidence interval for the mean response time. (d) (6 points) Name three factors that make the above confidence interval smaller. Stat 430 Name: Midterm Exam Oct 9, 2008 2. Google search response times - Parameter Estimation Assume that response times from Google are realizations of n = 55 independent random variables Xi (with i = 1, ..., 55) that all come from the Exponential distribution Expλ . (a) (15 points) Based on the data from the previous page, derive a maximum likelihood estimate λ̂ for the rate parameter λ. (b) You find out that in this particular study, the computer stopped recording values at 0.5 seconds and denoted this as a time out. During the study, this happened 5 times (making the ’real’ number of observations n = 60). i. (4 points) For random variable X ∼Expλ , what is the probability of a time out? ii. (3 points) How does the additional information affect the estimate for the rate λ? iii. (3+5 points) Setup a likelihood function that regards all 60 values (i.e. the 55 response times and the 5 time outs). (Bonus for a re-worked λ̂) Stat 430 Name: Midterm Exam Oct 9, 2008 3. Central Limit Theorem A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the time, and 51 pennies 10 percent of the time. (a) (4 points) Find the expected value and the variance for the amount that the bank loses on a typical roll. (b) (5 points) Explain why the amount of money that the bank loses on 100 rolls can be approximated by a normal distribution. Determine the parameters of the distribution. (c) (4 points) What is the (approximate) probability that the bank will lose at least 25 cents in 100 rolls? (d) (4 points) Find the (approximate) probability that the bank will lose exactly 25 cents in 100 rolls. Make sure that the continuous approximation is close to the discrete probability. (e) (2 points) Find the (approximate) probability that the bank will lose any money in 100 rolls. (f) (6 points) How many rolls does the bank need to collect to have a 99 percent chance of a net loss? Stat 430 Name: Midterm Exam Oct 9, 2008 4. Hits on a Web Site The number of hits on a popular Web page is distributed according to a Poisson distribution with arrival rate = 6 per minute. One begins observation at exactly noon. (a) (6 points) What is the probability of 4 or less hits in the first minute? Give a distribution and parameter(s). (b) (6 points) Compute the probability that the time till the first hit exceeds 10 seconds. Give a distribution and parameter(s). (c) (4 points) What is an appropriate distribution for the time until the 4th hit? State the distribution and parameter value(s). (d) (2 points) Evaluate the probability that the time till the 4th hit exceeds 24 seconds. (e) (2 points) Find the mean and the variance of the time till the 4th hit. (f) (5 points) The number of hits in the first hour is Poisson with mean 360. You would like to know the probability of more than 370 hits. Exact calculation isn’t really feasible. So approximate this probability and justify your approximation. Stat 430 Name: Midterm Exam You’re done! What is your overall score (without any bonus) on the exam? (out of 100 points) If you get the score right within ±3 points, you’ll get a bonus. Oct 9, 2008