Week Cardio Class Attendees Yoga Class Attendees Strength Training Class Attendees

$\backslash$table$[[1,45,30,35],[2,50,28,32],[3,48,32,36],[4,52,35,38],[5,47,31,33],[6,49,30,37],[7,51,33,34],[8,53,36,40],[9,54,29,39],[10,50,31,36],[11,55,34,38],[12,57,33,37],[13,53,30,39],[14,56,35,42],[15,52,32,37],[16,58,31,39],[17,60,29,41],[18,62,34,40],[19,59,36,43],[20,61,33,42],[21,64,35,44]]$A gym, $$FitLife Gym,$$ wants to forecast the number of attendees for weekly fitness classes. This forecast will help with class scheduling, staff management, and promotional efforts. The attached file shows the sales data for the past $21$ weeks.

$1.$ Construct a time series plot for the total number of attendees. Comment on the component$($s$)$ present in the data.

$2.$ Use a $4-$week moving average for this time series to develop a forecast for the total number of attendees during week $22.$

$3.$ Use exponential smoothing with a $=2$

for this time series to develop a forecast for the total number of attendees during week $22.$ Use the actual number of attendees as the forecasted value in week $1$ to get started.

$4.$ Compare the $4-$week moving average forecast with the exponential smoothing forecast with a $=2.$ Which provides the better forecast based on MAPE?