Curve Fitting, Regression, and Correlation in Software

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CHAPTER 8 Curve Fitting, Regression, and Correlation
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8.58. Table 8-27 shows the frequency distributions of the final grades of 100 students in mathematics and physics. With reference to this table determine (a) the number of students who received grades 70 through 79 in mathematics and 80 through 89 in physics, (b) the percentage of students with mathematics grades below 70, (c) the number of students who received a grade of 70 or more in physics and less than 80 in mathematics, (d) the percentage of students who passed at least one of the subjects assuming 60 to be the minimum passing grade. Table 8-27 MATHEMATICS GRADES 40 49 PHYSICS GRADES 90 99 80 89 70 79 60 69 50 59 40 49 TOTALS 1 3 3 7 4 6 5 15 1 5 9 6 4 25 23 20 10 50 59 60 69 70 79 2 4 10 5 2 80 89 4 6 8 2 90 99 4 5 1 TOTALS 10 16 24 21 17 12 100
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(a) Proceed down the column headed 70 79 (mathematics grade) to the row marked 80 89 (physics grade). The entry 4 gives the required number of students. (b) Total number of students with mathematics grades below 70 (number with grades 40 49) 7 15 25 47 47>100 47%. (number with grades 50 59) (number with grades 60 69)
Percentage of students with mathematics grades below 70
(c) The required number of students in the total of the entries in Table 8-28, which represents part of Table 8-27. Required number of students 1 5 2 4 10 22.
CHAPTER 8 Curve Fitting, Regression, and Correlation
Table 8-28 MATHEMATICS GRADES PHYSICS GRADES 60 69 PHYSICS GRADES 90 99 80 89 70 79 1 5 70 79 2 4 10
Table 8-29 MATHEMATICS GRADES 40 49 50 59 40 49 3 3 50 59 6 5
(d) Referring to Table 8-29, which is taken from Table 8-27, it is seen that the number of students with grades below 60 in both mathematics and physics is 3 3 6 5 17. Then the number of students with grades 60 or over in either physics or mathematics or both is 100 17 83, and the required percentage is 83>100 83%. Table 8-27 is sometimes called a bivariate frequency table or bivariate frequency distribution. Each square in the table is called a cell and corresponds to a pair of classes or class intervals. The number indicated in the cell is called the cell frequency. For example, in part (a) the number 4 is the frequency of the cell corresponding to the pair of class intervals 70 79 in mathematics and 80 89 in physics. The totals indicated in the last row and last column are called marginal totals or marginal frequencies. They correspond, respectively, to the class frequencies of the separate frequency distributions of mathematics and physics grades.
8.59. Show how to modify the formula of Problem 8.31 for the case of data grouped as in Table 8-27.
For grouped data, we can consider the various values of the variables x and y as coinciding with the class marks, while fx and fy are the corresponding class frequencies or marginal frequencies indicated in the last row and column of the bivariate frequency table. If we let f represent the various cell frequencies corresponding to the pairs of class marks (x, y), then we can replace the formula of Problem 8.31 by
n a fxy (1) r B Sn a fxx2
Q a fx xR Q a fy yR
Q a fxxR T Sn a fy y2
Q a fy yR T
If we let x x0 cxux and y y0 cyuy, where cx and cy are the class interval widths (assumed constant) and x0 and y0 are arbitrary class marks corresponding to the variables, the above formula becomes n a fux uy (2) r B Sn a fx u2 x Q a fx ux R Q a fy uy R
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