7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(Yn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.If X1, X2, ... , Xn constitute a random sample of size n from an exponential population, show that X is a consis-tent estimator of the parameter θ.Suppose that in a certain chemical process the reaction time (in hours) is related to the temperature (°F) in the chamber in which the reaction takes place, according to the simple linear regression model where β0 = 5.23, β1 = -0.01, and σ = 0.09. If the temperature is 260°F, what is the probability that the reaction time is between 2.51 and 2.7 hours? Suppose five observations are made independently on reaction time, each one for a temperature of 260°F. What is the probability that all five times are between 2.51 and 2.7 hours? If two independently observed reaction times for temperatures are 1° apart, what is the probability that the time at the higher temperature exceeds the time at the lower temperature?
- Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxyA dietitian wishes to see if a person’s cholesterol level will be changed if the diet is supplemented by a certain mineral. Four subjects were pre-tested, and they took the mineral supplement for a 6-week period. The results are shown in the table. Is there sufficient evidence to conclude that the population mean of cholesterol levels has been changed after six weeks at α=0.2α=0.2? Assume that the differences are from an approximately normally distributed population. Subject Cholestrol Level (mg/dl) Cholestrol Level after 6 Weeks (mg/dl) dd ¯dd¯ (d−¯d)2(d-d¯)2 1 206 217 11 2 219 184 -35 3 202 204 2 4 213 205 -8 Total -30 a) Calculate the mean, the sum of the squared deviation from the mean, and the standard deviation of differences. Do not include the unit for each answer: ¯d=d¯= (do not round) ∑(d−¯d)2=∑(d-d¯)2= (do not round) sd=sd= (rounded to one decimal place) b) Perform the hypothesis test in the following steps: Step 1.…The demand for a commodity is given by Q = β0 + β1P + u, whereQ denotes quantity, P denotes price, and u denotes factors other thanprice that determine demand. Supply for the commodity is given byQ = ϒ0 + ϒ1P + v, where v denotes factors other than price that determinesupply. Suppose that u and v both have a mean of zero, have variancesσ2u and σ2v, and are mutually uncorrelated.a. Solve the two simultaneous equations to show how Q and P dependon u and v.b. Derive the means of P and Q.c. Derive the variance of P, the variance of Q, and the covariancebetween Q and P.d. A random sample of observations of (Qi, Pi) is collected, and Qi isregressed on Pi. (That is, Qi is the regressand, and Pi is the regressor.)Suppose that the sample is very large. i. Use your answers to (b) and (c) to derive values of the regressioncoefficients. ii. A researcher uses the slope of this regression as an estimate of theslope of the demand function (β1). Is the estimated slope too largeor too small?
- An experiment is conducted to determine the relationship between the amount of a certain drug in the bloodstream and the length of time it takes to react to a stimulus. A random sample of 5 persons who took this drug is selected. The amount of drug in the bloodstream x and the reaction time y in selected persons showed that Amount of drugs x: 2 1 4 3 5 Reaction time y: 1 1 2 2 4 Σx=15 , Σx² = 55, Σy=10, Σy² = 26, Σxy = 37 (a) Compute SSxx, SSyy, and SSxy (b) Compute the correlation coefficient r and explain it in the context of the problem. (c) Find the equation of the regression line between y and x.Refer to Exercise 10 in Section 3.2. Assume that X = 30.0 ± 0.1 Pa, h = 10.0 ± 0.2 mm, and μ = 1.49 Pa · s with negligible uncertainty. a) Estimate V and find the uncertainty in the estimate. b) Which would provide a greater reduction in the uncertainty in V: reducing the uncertainty in τ to 0.01 Pa or reducing the uncertainty in h to 0.1 mm?In terms of the model parameters, state the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary.
- Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).2- An expert estimates that the distribution parameter for durability times of parts produced with machine A in the factory is different from the distribution parameter for durability times of parts produced with machine B. Durability times of 4 parts produced from machine A and 4 parts produced from machine B are given below. Find the Mann-Whitney U value by using these data. a) 18 B) 6 NS) 16 D) 20 TO) 12