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- You are conducting quality control for a company that manufactures LED displays. The factory you are assessing is supposed to have a manufacturing defect rate of 1 in 100 LED displays. As part of your assessment, you want to verify this defect rate by analyzing a random sample of LED displays. You are planning to randomly sample 1500 displays from this factory and observe how many of them contain manufacturing defects. Let Zi be equal to 1 if the i’th display has a defect and 0 otherwise, for i = 1,...,1500. (a) What is the statistic that you will use to estimate the defect rate for this factory? How do you compute it using Z1, Z2, . . . , Z1500? (b) Assuming that the true defect rate for this factory is in fact 1 in 100 displays, can we approximate the sampling distribution of the statistic that you selected in part (a) using a normal distribution? Please state and check the requirements for applying the approximation, and identify the mean and standard deviation of the normal…true or false. if you found X2 = 10 with df= 5 you would fail to reject Ho at the 5% significance level.In an effort to make the distribution of income more nearly equal, the government of a country passes a tax law that changes the Lorenz curve from y = 0.98x2.1for one year to y = 0.32x2 + 0.68x for the next year. Find the Gini coefficient of income for both years. (Round your answers to three decimal places.) after before
- Suppose you are interested in the causal effect of Xi on Yi and Yi = α + βXi + ϵi. Moreover, cov(Xi , ϵi) = 0. However, you do not observe Yi . Instead, you only observe a proxy for Yi . Denote this proxy Y'i and assume it is related to Yi as follows: Y'i = Yi + µi where cov(µi , ϵi) = cov(µi , Xi) = 0. Suppose you regress Y'i on Xi . Would the resulting coefficient β' provide an unbiased estimate of β?Let X1, . . . , Xn ∼ iid Unif(0, θ). In class, we compared the MLE estimator θˆ = X(n) and the unbiased estimator δ(x) = (n+1)/n X(n). (a) Find the MSE of the MOM estimator θ ̃ = 2X ̄. How does this compare to the unbiased estimator? (b) Consider the set of estimators for θ given by δa(x) = aX(n) where a > 0. We note that the MLE and its unbiased adjustment have this form with different values of a. Find a formula for the MSE of δa(x) as a function of the choice of a. (c) For what value of a, dose δa(x) minimize the MSE? Does this coincide with any of the estimators we have previously considered?A regression on the original regressors, ?̂t2 and a constant term yields the following statistics: R2 = 0.296041 F = 1.177507 coeff of ?̂t2 has a t-statistic of 2.876 With this information, which test can you implement to deal with the problem omitted variables and why? Implement the test as stated in b(i) and interpret the results. What is (are) the consequence(s) of the problem alluded to above on the estimators?
- See the attached image for the introduction. In terms of variables xi and parameters βi, write the null and alternative hypotheses for testing whether, after including Price/Square Feet(x2) in the model already, the further incorporation of the other 2 explanatory variables (x1, x3) adds any useful information for explaining pricey. Also, give the value of the F statistic and its degrees of freedom (df).1. A researcher reports an F-ratio with df(between) = 3 and df(within) = 28. How many treatment conditions were compared in the experiment? If samples have an equal number of participants, how many are in each treatment? 2. A researcher obtains an F = 4.10 with df = 2, 14. Is this value sufficient to reject the null with α = .05? Is it sufficient enough to reject the null with α = .01?- Let I = 1.135013 be the sample mean of an iid sample r1,..., x50 from a gamma population Gamma(1, 3). Here B > 0 is the unknown parameter of interest. Construct an approximate 95%-CI for B.
- The mean ±1 sd of ln [calcium intake (mg)] among 25 females, 12 to 14 years of age, below the poverty level is 6.56 ± 0.64. Similarly, the mean ± 1 sd of ln [calcium intake (mg)] among 40 females, 12 to 14 years of age, above the poverty level is 6.80 ± 0.76. 1. What is the p-value corresponding to your answer to Problem(A)?2. Compute a 95% CI for the difference in means between the two groups. the answer of problem(A) is: t= -1.1314Question 16 Match the p-values with the appropriate conclusion: I. 0.00001 II. 0.0297 III. 0.0721 IV. 0.4490(a) The evidence against the null hypothesis is significant, but only at the 10% level. (b) The evidence against the null and in favor of the alternative is very strong. (c) There is not enough evidence to reject the null hypothesis, even at the 10% level. (d) The result is significant at a 5% level but not at a 1% level.A certain brand of upright freezer is available in three different rated capacities: 16ft3, 18 ft3, and 20 ft3. Let X = the rated capacity of a freezer of this brand sold at acertain store. Suppose that X has pmfx 16 18 20p(x) .2 .5 3a. Compute E(X), E(X2), and V(X).b. If the price of a freezer having capacity X is 70X – 650, what is the expectedprice paid by the next customer to buy a freezer?c. What is the variance of the price paid by the next customer?d. Suppose that although the rated capacity of a freezer is X, the actual capacityis h(X) = X - .008X2. What is the expected actual capacity of the freezer purchasedby the next custom