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Suppose that a statistic U has a probability density function that is positive over the interval a ≤ u ≤ b and suppose that the density depends on a parameter θ that can range over the interval α1 ≤ θ ≤ α2. Suppose also that g(u) is continuous for u in the interval [a, b]. If E[g(U) | θ] = 0 for all θ in the interval [α1, α2] implies that g(u) is identically zero, then the family of density functions { fU (u | θ), α1 ≤ θ ≤ α2} is said to be complete. (All statistics that we employed in Section 9.5 have complete families of density functions.) Suppose that U is a sufficient statistic for θ, and g1(U) and g2(U) are both unbiased estimators of θ. Show that, if the family of density functions for U is complete, g1(U) must equal g2(U), and thus there is a unique function of U that is an unbiased estimator of θ. Coupled with the Rao-Blackwell theorem, the property of completeness of fU (u | θ), along with the sufficiency of U, assures us that there is a unique minimum-variance unbiased estimator (UMVUE) of θ.
Use confidence interval to determine the estimated margin of error, Then determine the sample mean.
when dragons on planet pern lay eggs the eggs are either green or yellow. the biologists have observed over the years
a sample of 30 cookies is taken to test the claim that each cookie contains an average of at least 9 chocolate chips.
Appalachian Coal mining believes it can enhance labor productivity and therefore net revenue by reducing air pollution in its mines. It estimates that the marginal cost function for reducing pollution by installing additional capital equipment is.
How did you think that your sample would compare to the national mean? This is your hypothesis. State either the alternate or null hypothesis and identify which one you are stating.
a. What is the sample proportion, p 5 proportion of contacts that resulted in a sale being made? b. For simple random samples of this size, what is the probability that p # 0.10?
body mass index bmi for newborns follows a normal distribution with a mean of 14 and standard deviation of .95. a
Create a bar chart with an appropriate vertical axis, horizontal axis, and title - what your final chart should look like, although your titles and numbers will be different. Remove my example and insert your own.
How would I go about finding the probabilities for the following discrete random variables? Among a shipment of 5,000 tires, 1,000 are slightly blemished. If one purchases 10 of these tires, what is the probability that 3 or less (P (x
Ethnic Group A: N = 15 M= 55 S2 = 6.5 Ethnic Group B : N = 23 M= 51 S2 = 4.5. If the standard deviation of the distribution of the difference between means is .76, what is the t score?
Analysis of the data shows a standard deviation of 0.61 minutes. At the 0.05 level, test whether the standard deviation of usage times for this bank's customers might actually be less than the amount predicted at the regional meeting.
Mazda 6s resulting from a 10-mph crash into a concrete wall, the research group tests six cars of this type and assesses the amount of damage. Describe the population and sample for this problem.
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