Consider the following regression Yi = βXi + ui. Show that (a) the OLS estimator of β is βˆ = (Σ(XiYi))/(Σ(Xi^2)) (b) (Σ(uˆi)/)n = 1/n(Σ(Yi − Yˆi))=/0
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Consider the following regression Yi = βXi + ui. Show that (a) the OLS estimator of β is βˆ = (Σ(XiYi))/(Σ(Xi^2)) (b) (Σ(uˆi)/)n = 1/n(Σ(Yi − Yˆi))=/0
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.The following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02also compute the regression equation in which you predict Y using X as the predictor variable
- Suppose that you have the following three regression models: Yi B₁ + B₂x₁ + Uj regression) Y₁ = a₁ + a₂x₁ + eį, if i is male male individuals) Yi = ₁ + ₂x₁ + Ei, if i is female (separate regression for female individuals) ESS RSS The error terms satisfies that E[ui] = E[ei] = E[ei] = 0. Then, we find the following RSS and ESS: Pooled Regression 124.3 25 (separate regression for (pooled 71.1 11.4 separate separate regression for regression for male individuals female individuals 56.6 8.6 The number of sample size is given by 18. Select the correct statement(s). The RSS from the pooled regression is always smaller than the sum of RSSS from the separate regression models. In the above setup, Chow test is equal to 2. If e; is correlated with €;, then the OLS estimator of a2 from the separate regression will be is biased. If the variances of e; is different from €;, then the OLS estimator of 3₂ is no longer unbiased estimator. If we add another variable z; to the pooled regression, then the…There is a linear regression: Yi = B0+ B1(Xi^2)+ ei present, where Xi is squared. ei ∼ N(0,σ2). How would I derive LSE for B0 and B1 and their variance?A particular article presented data on y = tar content (grains/100 ft³) of a gas stream as a function of x₁ = rotor speed (rev/min) and x₂ = gas inlet temperature (°F). The following regression model using X₁, X2, X3 = ×₂² and ×4 = X₁X₂ was suggested. (mean y value) = 86.5 – 0.121x₁ +5.07x2 - 0.0706x3 + 0.001x4 (a) According to this model, what is the mean y value (in grains/100 ft³) if x₁ = 3,400 and x₂ = 55. grains/100 ft³ (b) For this particular model, does it make sense to interpret the value of ₂ as the average change in tar content associated with a 1-degree increase in gas inlet temperature when rotor speed is held constant? Explain. Yes, since there are no other terms involving X2. O Yes, since there are other terms involving X₂. ● No, since there are other terms involving X2. O No, since there are no other terms involving X2.
- Researchers studying the association between (x)the number of daily hours of intense artificial lightand (y) the rate of plant growth (cm/month) foundsx = 1.5, sy = 3.0, and r = 0.80. The regression modelthey created predicts growth of 5 cm/month with 2 hoursof this light per day. What’s the predicted growth rate for3 hours of artificial light daily?A) 5.4 cm/moB) 5.8 cm/moC) 6.6 cm/moD) 7.0 cm/moE) 7.8 cm/moSuppose a logistic regression model is fitted for the probability of car ownership for residents of a certain city in Oman (Y=1 if a resident owns a car, Y=0 if a resident does not own a car). Suppose the explanatory variables used are x1=no. of years a resident spent in schooling and x2 is gender of the resident of the city (x2=1 for a male and x2=0 for a female resident) a) Interpret el and e82 b) if BO= -1.6, B1=0.4 and B2=3, estimate the probability of a resident in the city owning a car.Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature (°F) in the chamber in which the reaction takes place according to the simple linear regression model with equation y = 5.10 0.01x and a = 0.085. USE SALT (a) What is the expected change in reaction time for a 1°F increase in temperature? For a 9°F increase in temperature? 1°F increase hr hr 9°F increase (b) What is the expected reaction time when temperature is 220°F? When temperature is 260°F? 220°F hr ✓hr 260°F 2.5 (c) 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 2.37 and 2.63 hours? (Round your answer to four decimal places.) (d) What is the probability that two independently observed reaction times for temperatures 1° apart are such that the time at the higher temperature exceeds the time at the lower temperature? (Round your answer to four decimal places.) 0.4602 X You…
- Run a simple linear regressionin SPSS to know if previous experience (‘prevexp’: Previous Experience-months) significantly predicts current salary(‘salary’: Current Salary) in the work force . Use α =.05 Write the regression equation (use a and b provided in the Coefficients Table) Use this equation for: If a person has 112 months of previous experience, what is the person’s predicted current salary?A) A linear regression has a =6 and b=5 what is y predicted as when x=9? B) A linear regression has b=3 and a=4.What is the predicted Y for x=7?A regression was run to determine if there is a relationship between the happiness index (y) and life expectancy in years of a given country (x).The results of the regression were: y=a+bxa=-0.797b=0.181 (c) If a country increases its life expectancy, the happiness index will increase decrease (d) If the life expectancy is increased by 1 years in a certain country, how much will the happiness index change? Round to two decimal places.
