3. The Lévy distribution with parameter > 0 is characterized by having the density 2023 exp(-21), (b) (a) Suppose we have samples ₁,...,n from the Lévy distribution with parameter 8. Find the maximum likelihood esti- mator for 0. (c) f(x; 0) = (d) x>0 T≤0 Is your estimator & from part (a) unbiased? are samples from a Gamma distribution with parameters What is Var(). The hint from part (b) will be Hint: 1, k = 1/2 and X = 1/(20). useful here as well. Show that the distribution of the samples 1/11,...,1/n is in fact a Gamma distribution with the parameters spec- ified in the hint of part (b). It might be useful to know that I'(1/2) = √√√.

3) Suppose test scores are measured by the Gaussian Normal Distribution N(X,75,10) calculate the following. a) Pr(80 < X < 90) b) Pr(50 < X < 70) c) The 95 th percentile

1. In the past, a chemical company produced 880 pounds of a certain type of plastic per day. Now, using a newly developed and less expensive process, the mean daily yield of plastic for the first 50 days of production is 871 pounds; the standard deviation is 21 pounds. Do the data provide sufficient evidence to indicate that the mean daily yield for the new process is less than that of the old procedure? (Use α=0.05) (d) Conclusion of the test above is Reject the null hypothesis and the mean daily yield for the new process is less than that of the old procedure. Reject the null hypothesis and the mean daily yield for the new process is not less than that of the old procedure. Do not reject the null hypothesis and the mean daily yield for the new process is less than that of the old procedure. Do not reject the null hypothesis and the mean daily yield for the new process is not less than that of the old procedure.

Vehicle speeds at a certain highway location are believed to have approximately a normal distribution with mean µ = 50 mph and standard deviation σ = 5 mph. The speeds for a randomly selected sample of n = 25 vehicles will be recorded. (a) Give numerical values for the mean and standard deviation of the sampling distribution of possible sample means for randomly selected samples of n = 25 from the population of vehicle speeds. Mean = s.d.(x) = (b) Use the Empirical Rule to find values that fill in the blanks in the following sentence. For a random sample of n = 25 vehicles, there is about a 95% chance that the mean vehicle speed in the sample will be between and mph. (c) Sample speeds for a random sample of 25 vehicles are measured at this location, and the sample mean is 59 mph. Given the answer to part (b), explain whether this result is consistent with the belief that the mean speed at this location is μ = 50 mph. A sample mean of 59 mph (when n = 25) --Select--- be consistent with…