Provide ASAP:-
With reference to Exercise express 95% limits of prediction for the extraction efficiency in terms of the extraction time x0. Choosing suitable values of x0, sketch graphs of the loci of the upper and lower limits of prediction on the diagram of part (a) of Exercise. Note that since any two sets of limits of prediction obtained from these bands are dependent, they should be used only once for a single extraction time x0. Exercise A chemical company, wishing to study the effect of extraction time on the efficiency of an extraction operation, obtained the data shown in the following table: Extraction time Extraction (minutes) X efficiency (%) Y 27 57 45 64 41 80 19 46 35 62 39 72 19 52 49 77 15 57 31 68 (a) Draw a scatter plot to verify that a straight line will provide a good fit to the data, draw a straight line by eye, and use it to predict the extraction efficiency one can expect when the extraction time is 35 minutes. (b) Fit a straight line to the given data by the method of least squares and use it to predict the extraction efficiency one can expect when the extraction time is 35 minutes.\f\fWith reference to the preceding exercise, (a) construct a 95% confidence interval for 8, the elongation per thousand pounds of tensile stress; (b) find 95% limits of prediction for the elongation of a specimen with x = 3.5 thousand pounds. Exercise In the accompanying table, x is the tensile force applied to a steel specimen in thousands of pounds, and y is the resulting elongation in thousandths of an inch: X 2 3 4 5 6 y 14 33 40 63 76 85 (a) Graph the data to verify that it is reasonable to assume that the regression of Y on x is linear. (b) Find the equation of the least squares line, and use it to predict the elongation when the tensile force is 3.5 thousand pounds.With reference to Exercise find (a) a 95% confidence interval for the average number of cars inspected in the given period of time by a person who has been working at the inspection station for 8 weeks; (b) 95 % limits of prediction for the number of cars that will be inspected in the given period of time by a person who has worked at the inspection station for 8 weeks. Exercise The following table shows how many weeks a sample of 6 persons have worked at an automobile inspection station and the number of cars each one inspected between noon and 2 P.M. on a given day: Number of weeks employed X| Number of cars inspected Y 2 13 7 21 23 14 5 15 12 21 (a) Find the equation of the least squares line which will enable us to predict y in terms of x. (b) Use the result of part (a) to estimate how many cars someone who has been working at the inspection station for & weeks can be expected to inspect during the given 2-hour period.\fThe decomposition of the sums of squares into a contribution due to error and a contribution due to regression underlies the least squares analysis. Consider the identity M - y - (3 - y) = (X - >) Note that H=atbx = ] - bx+ bx = y+ b(x - X) 80 # - J = b(x -x) Using this last expression, then the definition of b and again the last expression, we see that and the sum of squares about the mean can be decomposed as total sum of squares error sum of squares regression rim of squares Generally, we find the straight-line fit acceptable if the ratio regression sum of squares =1 - total sum of squares is near 1. Calculate the decomposition of the sum of squares and calculate r2 using the observations in Exercise. Exercise The following data pertain to the number of computer jobs per day and the central processing unit (CPU) time required. Number of jobs X| CPU time Y N 2 5 4 4 9 5 10 (a) Use the first set of expressions on page 304, involving the deviations from the mean, to obtain a least squares fit of a line to the observations on CPU time. (b) Use the equation of the least squares line to estimate the mean CPU time at x = 3.5.With reference to the preceding exercise, change the equation obtained in part (a) to the form " alecx and use the result to rework part (b). Exercise The following data pertain to the cosmic ray doses measured at various altitudes: Altitude (feet) X| Dose rate (mem/year) Y 50 28 450 30 780 32 1,200 36 4,400 51 4,800 58 5,300 69 (a) Fit an exponential curve (b) Use the result obtained in part (a) to estimate the mean dose at an altitude of 3,000 feet
