 Hello and welcome to the session. In this session, we will discuss a question which says that the following table shows the average monthly production of bikes in thousands for the years 1997 to 2006. Then for this data, determine first half firm yearly moving average figures, second part firm yearly central moving average, also plant the graph for actual values and trend values. Now before starting the solution of this question, we should know a result. And that is the procedure for calculating firm yearly moving averages. Now for calculating the firm yearly moving averages in the question, a data will be given to us. We will consider this data in which the various values for the particular years are given to us. So there, by the use of the data given, we will find out firm yearly moving total and firm yearly moving average then firm yearly moving average centred. Now the first step to calculate firm yearly moving total is firm years, that is, we will add these firm values and then in between these four values, this data in between the second and third year, that is, at this place. So we write A plus B plus C plus D and let us denote it. That is, we will leave this value, that is, the values for the second, third, fourth and fifth year between third and fourth year, that is, we will write here B plus C plus D plus E and let us calculate it for yearly moving total. Now for calculating moving totals, that is, A to A3 and A4 by A4 and in this case A4 by 4 and 3 and this is and divide by centred and place this value as we will write M1 B2 at this place. It will be M2 by upon 2 to this process we average the points, that is, the third year and B1 then fourth year and B2 to get the trend line. That is a key idea. Now in the question, this data is given to us at moving average. Now we will use the result, which is given in the key idea and we will find material for the given data. For the given data, we have made a table in which the first column is representing the years the average monthly production of bikes in thousands. Then in the next column, we have to find the four yearly moving totals. Now these are the first fundamental in between the values that is 35.5 which I am calculating is equal to 157. Now for the four yearly moving average we will divide this total by 4 so this will be 157 by 4 which is equal to 39.25 that is this value 31.0 46.2 which I am calculating is equal to 167.7 we will divide this total by 4 so it will be 167.7 divided by 4 9 to 5 Now in the next step, we will leave this value in between these two total by 4 so it will be 180.2 by 4 which is equal to 35.05 so these four values is 189.2 For the four yearly moving average we will divide this total by 4 and on the right hand side 4 total in between these two years these four values if we give 197 now on the right hand side 4 it will be 49.425 and put the total in between these two years by 4 years Now in the next step, we will add total in between these two years which it will give 193.5 and 193.5 divided by 4 is 48.375 We have continued this process until the last year averages and divide by 2 by which is the year 1999 that is equal to 14.3 and divide by 2 and place it against this year which is the year 2000 two moving averages and dividing by 2 we are getting 43.487 right it as 43.5 these two moving averages and divide them with which is the year 2001 so whereby adding them and dividing them by 2 we are getting the result as now for the next we will add these two values and divide them by 2 moving averages and divide them by 2 which against this year we will add these two moving averages and divide them by 2 against this year 0.8125 or you can say 49.8 here these are the actual values so now we will plot the graph for the actual and trend values this is represented the average monthly production of y's in thousands if on the graph that is the point 1998 37.5 1998 37.0 on the graph we will plot this point which is the point 1999 41.0 in 1999 41.0 on the graph we will plot so we have plotted all these points on the graph hence we are getting the trend values now we will plot the points 43.5 then 2048.4 2003 which is so this is the required points which are discussed earlier on the trend line so this is the solution of the given question and that's all for this session hope you all have enjoyed this session