Concept explainers
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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.arrow_forwardAssume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .arrow_forwardShow that the mean of a random sample of size n from an exponential population is a minimum variance unbiased estimator of the parameter 0.arrow_forward
- Let X1,...,Xn be a sample from normal with mean theta and variance 1. Construct an unbiased estimator of theta^2 based from X^2. Find its variance and compare it with the Cramer-Rao lower bound.arrow_forwardLet X1, ..., Xn be a sample from an exponential population with parameter λ.(a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased?(c) Is the estimator consistent?arrow_forwardThe number of accidents in a city follows a Poisson process with a mean of 2 per day and the number X; of people involved in the ith accident has the distribution (independent) P{X; = k} = (k2 1). Find the mean and variance of the number %3D 2* of people involved in accidents per week.arrow_forward
- Q.8). In a regression calculation, a researcher finds that the explanatory variable x has mean 100 and SD 10, and the response variable y has mean 250 and SD 40. The regression equation is found to be y^ = 450 and y? 2x. What is the correlation between x (a) cannot tell from the information available (b) -0.8 (c) -0.5 (d) 0.4 (e) 0.1 Q.10) a) What are the differences between the data and information? b) Explain the differences between the types of Statisticsarrow_forwardA random sample of n = 19 winter days in Denver gave a sample mean pollution index x1 = 43. Previous studies show that o1 = 10. For Englewood (a suburb of Denver), a random sample of n2 = 18 winter days gave a sample mean pollution index of x2 = 34. Previous studies show that o2 = 13. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. O Ho: H1 H2 O Ho: H1 = l2; H1: H1 < µ2 (b) What sampling distribution will you use? What assumptions are you making? O The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's…arrow_forwardFor an experiment it is randomly collected 40 number of observations and it is not known ex-ante whether this data is normally distributed or not. For this experiment the individual monthly salaries are summed in U.S. dollar terms and found to be 20.000S. For these 40 number of observations the variance term is found to be equal to 6400. Given that set of information, a. Please discuss briefly whether the central limit theorem is appropriate to compare or analyze the mean value of this sample within certain intervals; b. Please calculate and interpret the probability that the sample mean will be within 1.5*standard deviation of the mean; c. Please write down your own question along with an explicit solution related to the distribution of the sample mean. Here the constraint is that you need to calculate an interval which corresponds to some probability between 0.05 and 0.20 of this distribution of the sample mean.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,