Consider a signal detection problem involving two hypotheses: H₁ : X = W and where H₁ : X = 8 + W, Ton || is a known signal vector, and W is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is P(H₁) = P(H₁) = 0.5. Suppose we observe X, but based on X we still determine that the two hypotheses are equally likely. What must be true about X?
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- 2) Let G and H be two independent unbiased estimators of 0. Assume that the variance of G is two times the variance of H. Find the constants a and b so that aG + bH is an unbiased estimator with the smallest possible variance for such a linear combination.Consider a signal detection problem involving two hypotheses: Ho : X = W and where H₁ : X = 8 + W, Suppose we observe is a known signal vector, and Ŵ is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is, P(Ho) = P(H₁) = 1 0 S = -1 0 1 Which hypothesis is more likely? 0.63 0.21 X = -1.19 0.84 -0.04A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5−9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L.8.31 What are the appropriate null and alternative hypotheses to compare the means of the two groups?8.32 What is the appropriate test procedure for the hypotheses in Problem 8.31?8.33 Carry out the test in Problem 8.32 using the criticalvalue method.8.34 Provide a 95% CI for the true mean difference in FEV between 5- to 9-year-old children whose parents smoke and comparable children whose parents do not smoke.8.35 Assuming this is regarded as a pilot study, how many children…
- A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5-9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L.*8.31 What are the appropriate null and alternative hypotheses to compare the means of the two groups? *8.32 What is the appropriate test procedure for the hypotheses in Problem 8.31? *8.33 Carry out the test in Problem 8.32 using the criticalvalue method. *8.34 Provide a 95% CI for the true mean difference in FEV between 5- to 9-year-old children whose parents smoke and comparable children whose parents do not smoke. *8.35 Assuming this is regarded as a pilot study, how many…A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5−9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L. A) What are the appropriate null and alternative hypotheses to compare the means of the two groups? B) What is the appropriate test procedure for the hypotheses in Problem A? C) Carry out the test in Problem B using the criticalvalue method. D) Provide a 95% CI for the true mean difference in FEV between 5- to 9-year-old children whose parents smoke and comparable children whose parents do not smoke.A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5−9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L. A) Assuming this is regarded as a pilot study, how many children are needed in each group (assuming equal numbers in each group) to have a 95% chance of detecting a significant difference using a two-sided test with α = .05? B) Answer the question in Problem A if the investigators use a one-sided rather than a two-sided test. Suppose 40 children, both of whose parents smoke, and 50 children, neither of whose parents smoke, are recruited for the study. C) How much power would…
- A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5−9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L. Suppose 40 children, both of whose parents smoke, and50 children, neither of whose parents smoke, are recruitedfor the study.*8.37 How much power would such a study have using atwo-sided test with significance level = .05, assuming thatthe estimates of the population parameters in the pilot studyare correct?*8.38 Answer Problem 8.37 assuming a one-sided ratherthan a two-sided test is used.[0] A transmitter using 3-fold repetition coding uses the vector to represent a 0 bit and A to send a 1 bit. (Suppose a priori that a 0 bit and a 1 bit are equally likely.) Over the communication channel the transmitted vector is added to a vector consisting of independent Gaussian random variables with mean 0 and [1.21] variance 2. Suppose the receiver receives 0.18 Which bit is more likely to have 0.71 been transmitted?Consider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choice
- Suppose we are interested in the effect of gender on income and we estimate a model in which income is the dependent variable and the regressor is a dummy variable that takes the value 1 if the person is a man and O if the person is a woman. If we instead estimate a model that includes a dummy variable for women (takes value 1 if woman and 0 if man), then The fit of the model will change. We will get a different conclusion for testing a null hypothesis that there is no gender effect. The coefficient on the dummy variables will differ across models, but the estimated difference of mean income between men and women will not change. The coefficient on the dummy variables will be exactly the same.Suppose that y(x1,x2) = x1/x2, where x1 and x2 are two independent random variables. Which of the following equations is CORRECT?a production line is producing light bulbs. 90% of the bulbs are good and .10% are defective. Suppose each bulbls condition is dependent of the other bulbs condition Select three buls at randon: 1. P(all 3 are good)=? 2. P( 2 of 3 are good)= 3. P(1 of 3 are good)=
