Applied Statistics for Business & Economics

Student ID: Name: STA 103 – Homework You will write your answers below (showing all necessary work), then upload a scan of your homework to Gradescope. 1. The marketing manager of a company has noted that she usually receives 10 complaint calls from customers during a week (assume a week has 7 days), and that the calls are independent, and assume a Poisson distribution. (a) Find the probability that she receives exactly 5 complaint calls in one week. (b) Find the probability that she receives at least 2 complaint calls in one day. (c) Find the expected number of complaint calls in one month (assume a month has 30 days). 1 2. During the summer months of June, July, and August, an average of 5 marriages per month take place in a small city. Assuming that these marriages occur randomly and independently of one another, and assuming a Poisson distribution, find the following: (a) The probability that 4 marriages will occur in June. (b) The probability that between 14 and 16 marriages (inclusive) will occur over the 3 months of June, July, and August. (c) The probability that at at least 1 marriage will happen in a week of June (assume a month has 30 days). 2 3. An inspector for air pollution decides to inspect 8 of a company’s 16 trucks. 5 of those trucks have pollutants above the accepted level. Assume the trucks the inspector looked at were selected randomly. (a) What is the probability that he finds exactly 3 trucks with pollutants above the accepted level? (b) What is the probability that he finds at least 3 of the trucks with pollutants above the accepted level? (c) What is the expected number of trucks with pollutants above the accepted level he will find? 3 4. An I.R.S auditor will inspect 5 of 12 tax returns, out of which 6 contain illegitimate deductions. Assume he selects the returns randomly. (a) Find the probability that he catches exactly 3 illegitimate deductions. (b) Find the probability that he catches at least 2 illegitimate deductions. (c) Find the standard deviation of the number of illegitimate deductions he will catch. 4 5. For a standard normal random variable (i.e, for a Z ∼ N (µ = 0, σ = 1) find the following: (a) P (Z < 1.2) (b) P (Z > −2.3) (c) P (−1 < Z < 0.31) (d) The 97.5th percentile of Z (the Z value which has 0.025 in the upper tail). 5 6. Suppose the length of time students take in writing a standard entrance examination is normally distributed with mean 60 minutes, std. deviation 8 minutes. (a) Find the probability that a randomly selected student takes between 60 and 70 minutes to write the exam. Final Numerical Answer Below (b) Find the probability that a randomly selected student takes at most 80 minutes to write the exam. Final Numerical Answer Below (c) If a randomly selected student has taken over 40 minutes, find the probability they will take at most 80 minutes to write the exam. Final Numerical Answer Below (d) Find the 50th percentile for the time it takes for students to write the exam. Final Numerical Answer Below 6 7. A venture capital company feels that the rate of return (X) on a proposed investment is approximately normally distributed with mean 30% and standard deviation 10%. (a) Find the probability that the return will exceed 55%. Final Numerical Answer Below (b) Find the probability that the return will be less than 25% Final Numerical Answer Below (c) What is the expected value of the return? Final Numerical Answer Below (d) Find the 75th percentile of returns. Final Numerical Answer Below 7 8. A firm’s marketing manager believes that total sales for the firm next year can be modeled by using a normal distribution with mean 2.5 million and a standard deviation of $300,000. (a) What is the probability that the firm’s sales will exceed 3 million? Final Numerical Answer Below (b) What is the probability that the firm’s sales will fall within $150,000 of the expected level of sales? Final Numerical Answer Below (c) In order to cover fixed costs, the firm’s sales must exceed the break-event level of 1.8 million. What is the probability that sales will exceed the break even level? Final Numerical Answer Below (d) Determine the sales level that has only a 9% chance of being exceeded next year. Final Numerical Answer Below 8