• You are planning to invest in a high-tech company, and figure your rate of return over the coming year as follows: (where 100% says you doubled your money, -50% says you lost half etc.) Rate of return, X Probability, P(X=x) xP(X=x) x^2P(X=x) 100% 0.25 25 2500 50% 0.40 20 1000 0% 0.20 0 0 -50% 0.15 -7.5 375 Total 1 37.5 3875 a) Find the mean rate of return. b) Find the standard deviation of the rate of return. c) What does the standard deviation represent? d) Find the probability that you will earn more than 40%. 2. You have a factory where machines are producing at a mean daily rate of 100 products with a standard deviation of 10. You may assume a Normal distribution. a) Find the probability that one machine will produce more than 102 products tomorrow. b) Consider a random sample of 36 machines. Find the probability that the average daily production will be more than 102 products tomorrow. c) Why are the probabilities computed in a) and b) different? 3. A study was conducted by your company, with 324 random interested customers indicating that they would pay an average of $18.14 for your new product. The standard deviation was $2.98. a) Find the 95% confidence interval for the average that consumers would be willing to pay in the population. b) Find the 99% confidence interval for the average that consumers would be willing to pay in the population. c) Which interval (95% or 99%) is more likely to include the unknown population mean? Why? d) Test the hypothesis that interested consumers would be willing, on average, to pay $20.00 for the new product, against the two-sided alternative. (At the 5% level of significance). 4. Active consumers make up 14.6% of the market and spend an average of $16.23 per month on your product. Passive consumers make up 24.8% of the market and spend an average of $9.85 per month. The remaining consumers have an average spending of $14.77. Find the average spending for all consumers. 5. Analyze the Chapter 5 Case “Should We Keep or Get Rid of This Supplier” given on page 128 of your textbook. Answer the three discussion questions and limit your analysis, including any statistical methods and graphs, to one page. Are Kellerman’s calculations correct? These are the first items to verify. Take a close look at the data using appropriate statistical methods. Are Kellerman’s conclusions correct? If so, why do you think so? If not, why not and what should be done instead?