What is P(X = 3|Y=2)? %3D
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- Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?Suppose that the variable under consideration is normally distributed on each of the two populations and that you are going to use independent simple random samples to compare the population means. Fill in the blank and explain your answer: Unless you are quite sure that the ________ are equal, the nonpooled t-procedures should be used instead of the pooled t-procedures.If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.
- If X1, X2, and X3 constitute a random sample of sizen = 3 from a normal population with the mean μ and thevariance σ2, find the efficiency of X1 + 2X2 + X34relative toX1 + X2 + X33as estimates of μ.Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]A researcher is using a two-tailed hypothesis test with α = 0.05 to evaluate the effect of a treatment. If the boundaries for the critical region are t = ± 2.080, then how many individuals are in the sample?
- If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.If X is distributed as normal with μ = 20 and σ2 = 40 and a random sample of 16 variates is drawn from the population, what is the probability that their sample mean will be between 17 and 2?Suppose the random variables X and Y are inde lent and identically distributed. Let Z =aX +Y. If the correlation coefficient between X and Z is1/3 , then what is the value of the constant a ?
- A large manufacturing company investigated the service it received from its suppliers and discovered that, in the past, 38% of all material shipments were received late. However, the company recently installed a just-in-time system in which suppliers are linked more closely to the manufacturing process. A random sample of 150 deliveries since the just-in-time system was installed reveals that 33 deliveries were late. If we want to test whether the proportion of late deliveries was reduced signicantly at = 0:10 the null and alternative hypotheses are a. Null hypothesis (H0) b. Alternative hypothesis (HA)A major credit card company is interested in the proportion of individuals who use a competitor’s credit card. Their null hypothesis is H0: p=0.65H0: p=0.65, and based on a sample they find a sample proportion of 0.70 and a pp-value of 0.053. Is there convincing statistical evidence at the 0.05 level of significance that the true proportion of individuals who use the competitor’s card is actually greater than 0.65 ? Yes, because the sample proportion 0.70 is greater than the hypothesized proportion 0.65. A Yes, because the pp-value 0.053 is greater than the significance level 0.05. B No, because the sample proportion 0.70 is greater than the hypothesized proportion 0.65. C No, since the sample proportion 0.70 is exactly 0.05 away from the hypothesized proportion 0.65. D No, because the pp-value 0.053 is greater than the significance level 0.05.1)Remove the four potential outliers of 0, 0, 8, and 20, and then obtain a new histogram without the outliers. Does the data appear to be normally distributed now? 2)Assuming that the four potential outliers of 0, 0, 8, and 20 are not recording errors, repeat the hypothesis test from part (c) (again setting up the hypothesis test and using either the critical value or p-value approach), and compare your results with that obtained in (c). Did you make a different conclusion? 3)Imagine you know have to make a recommendation/conclusion to the company that hired you: Assuming that the four potential outlies are not recording errors, and looking at the two results above, would you recommend using the first test with the outliers or the second test with the outliers removed? There is no right or wrong answer here, I am interested in what you think and your reasoning.
