Concept explainers
For the simple linear regression model Y = β0 + β1x + ε with E(ε) = 0 and V(ε) = σ2, use the expression for
For what value of x∗ does the confidence interval for E(Y) achieve its minimum length?
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Mathematical Statistics with Applications
- 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 19.02arrow_forwardSome non-linear regressions can also be estimated using a linear regression model (using 'linearization'). Assume that the data below show the selling prices y (in dollars) of a certain equipment against its age x (in years). We'd like to fit a non-linear regression in the form y = cd* to estimate parameters c and d from the data by linearizing the model through In y In c+ (In d)x = b, + b, x. y y 6381 3 5394 5673 4980 2 5740 4896 (Click the button to copy or download the data.) Using Excel ot other software, the non-linear regression model y = cd can be estimated as: y = D*. (Round c and d to four decimal places, inlcuding any zeros.)arrow_forwardSuppose 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 = 5.40 -0.01x and a=0.08. USE SALT (a) What is the expected change in reaction time for a 1°F increase in temperature? For an 8°F increase in temperature? 1°F increase hr hr 8°F increasearrow_forward
- Prove OLS Estimator are independent of each other for the Multiple Linear Regression modelarrow_forwardSuppose 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…arrow_forwardFor the linear regression model Y = bo + b1(X): The p-value for the intercept is large: about 0.98 The p-value for the slope is very small: less than 2 times 10^(-16) What can we conclude? Since the p-value for the intercept is large, we can conclude that there is not a strong correlation between X and Y. Since the p-value for the intercept is large, we can conclude that there is a very strong correlation between X and Y. Since the p-value for the slope is very small, we can conclude that there is a very weak correlation between X and Y. Since the p-value for the slope is very small, we can conclude that there is a very strong correlation between X and Y. We are not able to assess the strength of the correlation between X and Y with the output provided.arrow_forward
- when the regression line passes through the origin thenarrow_forwardRun a simple linear regression in 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)arrow_forwardThe head of the finance department of Supermarket Company in White Exited, New York City prepared the following table. Values are in thousandpesos. 1.) Plot the Scatter Diagram 2.) Solve and plot for the linear regression equation ( y^= a + bx) on the same graph in a. b=nΣ xy-(Σ x)(Σ y)/n(Σ x2)-(Σ x)2 a=Σ y-bΣ x/n Sales y Cost Radio Advertising 76 10 85 12 80 11 55 6 57 9arrow_forward
- We wish to predict the salary for baseball players (y) using the variables RBI (x1) and HR (x2), then we use a regression equation of the form ˆy=b0+b1x1+b2x2y^=b0+b1x1+b2x2. HR - Home runs - hits on which the batter successfully touched all four bases, without the contribution of a fielding error. RBI - Run batted in - number of runners who scored due to a batters's action, except when batter grounded into double play or reached on an error Salary is in millions of dollars. The following is a chart of baseball players' salaries and statistics from 2016. Player Name RBI's HR's Salary (in millions) Adrian Beltre 104 32 18.000 Justin Smoak 34 14 3.900 Jean Segura 64 20 2.600 Justin Upton 87 31 22.125 Brandon Crawford 84 12 6.000 Curtis Granderson 59 30 16.000 Aaron Hill 38 10 12.000 Miquel Cabrera 108 38 28.050 Adrian Gonzalez 90 18 21.857 Jacoby Ellsbury 56 9 21.143 Mark Teixeira 44 15 23.125 Albert Pujols 119 31 25.000 Matt Wieters 66 17 15.800 Logan…arrow_forwardConsider the multiple regression model Y₁ = Bo + B₁x1₁j + B₂x2j+B3 x 3j+ €j under the usual assumptions labelled A1, A2, A3, A4, A5, A6. Briefly explain which type of graphs are performed in the analysis of residuals.arrow_forwardThe authors of the paper "Power-Load Prediction Based on Multiple Linear Regression Model"t were interested in predicting the load on the electric power system in China using data on y = Power consumption (in hundreds of millions of kwh), x, Population (in millions), and x, = Gross domestic %3D product (in billions of dollars), for 21 years. The model equation proposed in the paper is y = -113,527 + 0.974x, + 0.057x, + e. (a) According to this model, what is the mean power consumption (in hundreds of millions of kwh) for a year if the population was 160,000 million and the gross domestic product was 600,000 billion dollars? hundreds of millions of kwh (b) Interpret the value of B, in this model. When the [ gross domestic product v is fixed, the mean increase in [power consumption (in hundreds of millions of kwh) V associated with a 1-million unit increase in [population is 0.974.arrow_forward
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