Let a, b, and c be any real numbers and let x = (x1,x2 ...,Xị, .,x,) be a data set with a sample mean z and Let y = (Y1,½ --. \y »n) be a data set with a sample mean y. Let z = ax + by +c such that z = (ax, + by, + c, ax, + by,+ .,ax; + by; + c,.., ax, + by, + c). Show that z = až + bỹ + c.
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- The Department of Mathematics of a certain university in a recent survey was found to consist of 80 faculty members. The survey also revealed the information that among them, there are: 40 male professors; 38 married professors; 26 professors with more than 10 years of service; 15 married male professors; 14 married professors with more than 10 years of service; 13 male professors with more than 10 years of service; and 6 married male professors with more than 10 years of service. How many unmarried female professors have less than or equal to 10 years of service?Let $X$ and $Y$ be random variables with the space consisting of the four points (0,0),(1,1),(1,0),(1,-1) . Assign positive probabilities to these four points so that the correlation coefficient is equal to zero. Are $X$ and $Y$ independent?A real estate agent working in a large city believes that, for three-bedroom houses, the selling price of the house decreases by approximately $2,000 for every mile increase in the distance of the house from the city center. To investigate the belief, the agent obtained a random sample of 20 three-bedroom houses that sold in the last year. The selling price, in thousands of dollars, and the distance from the city center, in miles, for each of the 20 houses are shown in the scatterplot. The table shows computer output from a regression analysis of the data. Predictor Coef SE Coef T P Constant 301.7 1.80 167.17 0.000 Distance −2.158 0.149 −14.45 0.000 S=4.4336 R-sq=92.1% (a) Assume all conditions for inference are met. Construct and interpret a 95 percent confidence interval for the slope of the least-squares regression line. (b) Does the confidence interval contradict the agent’s belief about the relationship between selling price and distance from the city…
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