 Hello and welcome to the session. In this session, we will discuss a question which says that a factory gives in service training to its workers, which is followed by a test. It is considering whether it should terminate the services of any worker who does not do well in the test. The following data gives the test scores and productivity made by 10 workers during a certain period. And the SATA is given to us in which the test scores are given as 14, 16, 19, 21, 24, 25, 27, 30. And the corresponding productivity in terms is given as 11, 13, 17, 24, 28, 30, 32, 37. Telling us the coefficient of correlation between the test scores and the productivity, does it indicate that the termination of services of workers of low test score is justified? If the value was minimum productivity value of 100 units, what is the minimum test score that will ensure continuation of service? Now before starting the solution of this question, we should know some results. And that is the regression equation of x on y is given as x minus x bar is equal to b x y into y minus y bar the whole. Here, x bar is the mean value of x, y bar is the mean value of y and b x y is the regression coefficient of x on y. And the regression coefficient of x on y that is b x y is given by the formula summation d x d y over summation d y squared. Where d x is equal to x minus x bar and d y is equal to y minus y bar. That is d x and d y are the relations of the variables x and y from the arithmetic means of the series. And r is equal to summation of d x into d y over summation d x squared into summation d y squared. They are relation coefficient. Now these results will work out as a key idea to answer this question. And now we will start with the solution. First, we will make the table for the given data. That is, we will make the table for the test scores and pt index. Now let pt be denoted by pt table for the test scores and the productivity index. So we have drawn a table for the given data. In this, in the first column we have written the scores which are represented by x and in the second column we have written the productivity index that is represented by y. That means that if the test score of a worker is 14 then the productivity is 110 units and if the test score of a worker is 16 then the productivity is 130 units. The next column we will find d x which is equal to x minus x bar then in the next column we will find d y which is equal to y minus y bar. In the next column we will find d x squared then d y squared then in the last column we will find d x into d x. We are getting summation x is equal to 176 and on every value of y we are getting summation y is equal to 192. Now d is equal to summation x by n. Now here summation x is 170 number of observations which are 1, 2, 3, 4. So it will be which is equal to 22. The mean value of y is y bar which is equal to summation y over n which is equal to now summation y is 192. So this will be 24. This is 22. So it will be x minus 22 which will be y minus y bar is 24. So it will be y minus. Now we will find d x. Let us we will subtract 22 from the different values of x. Now here x is 20. So 40 minus 22 is minus 8 then 16 minus 22 is minus 6 19 minus 22 is minus 3 then 24 minus 22 is 2 25 minus 22 will give which is equal to y minus 24. So d y will be equal to 11 minus 24 which is minus 13 then 13 minus 24 which is minus 11 then 17 minus 24 is minus 7 24 minus 24 is 0 now d x minus 24 is 4 13 minus 24 is 6 32 minus 24 is we will find d x square that is we will square the different values of d x. So minus 8 square is 6 is 36 minus 3 square is 9 minus 1 square is 1 2 square is 4 3 square is 9 5 square is 25 we will square different values of d y. minus 13 square is 169 this is 121 49 0 square is 0 4 square is 6 d x into d y that is we will multiply different values of d x with different values of d y. Now minus 8 into minus 13 is 104 minus 6 into minus 11 is 7 is 21 minus 1 into 0 is 0 2 into 4 is 3 into 6 13 is 104 values of d x square we are getting summation d x square is equal to 212 then on adding different values of d y square we are getting summation d y square is equal to values of d x into d y we are getting summation of d x into d y is equal to 361 to calculate the coefficient of correlation between the test scores and the productivity and then we have to check indicate that the termination of services of workers of the load test score is justified or not. now using this result which is given in the d idea so the correlation coefficient equation d x d y is equal to 12 summation d y square is 624 and summation of d x into d y is 361 so putting all these values here this implies it further implies to 0.99 now we will get the value of 1 minus 1 now we are getting r is equal to 0.9925 and the maximum value of r is 1 that's the correlation between the scores and the productivity this means that the worker with the load score is less productive and the worker with the highest score is highly productive of load test scores is justified therefore the proposal productivity value of 100 year and the minimum test score ensure continuation of service now given r is equal to that is y is equal to 10 indicates that the minimum productivity value is 100 year now we have the continuation of service when the minimum productivity value is given as 100 years i is equal to 10 we have to find the value of x now we will get the regression line of x and y with the best estimated value of x for the given value of y the regression line is equal to b x y into y minus y bar the whole which implies r is equal to now b x y is summation of dx into dy over summation dy square into y minus y bar the whole x bar is 22 y bar is 24 into dy is 361 x 24 using all these values here this implies x minus 22 is equal to y minus 24th the whole other implies x minus 22 is equal to into y minus 24th the whole is 22 is equal to 0.8 y minus which further implies 0.578 y is the regression equation now we know that regression equation of x and y gives the value of x for a given value of y now let this be equation a now put in while in equation a we get 0.578 into 10 plus 8.128 is 908 to the nearest real number it will be a minimum of 0 and sure equation and that's all for this session