A company that produces wakeboards forecasts monthly sales S...

A company that produces wakeboards forecasts monthly sales $S$ during a two-year period to be $$S=2.7+0.142 t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)$$ where $S$ is measured in hundreds of units and $t$ is the time (in months), with $t=1$ corresponding to January $2014 .$ Estimate sales for each month. (a) January 2014 (b) February 2015 (c) May 2014 (d) June 2015

A company that produces wakeboards forecasts monthly sales $S$ during a two-year period to be
$$S=2.7+0.142 t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)$$
where $S$ is measured in hundreds of units and $t$ is the time (in months), with $t=1$ corresponding to January $2014 .$ Estimate sales for each month.
(a) January 2014
(b) February 2015
(c) May 2014
(d) June 2015

Key Concepts

Seasonal Component

The seasonal component captures the periodic fluctuations or cycles in the data that repeat at regular intervals, such as months or quarters. This cyclic pattern is often modeled using sinusoidal functions, which effectively describe the oscillatory behavior in the observed data and allow for adjustments in predictions based on recurring seasonal effects.

Time Series Forecasting

Time series forecasting is the process of using historical data to predict future values by identifying patterns, trends, and seasonal variations within the data. It involves developing mathematical models or algorithms to capture the underlying structure of the data over time, allowing for more accurate predictions of future observations.

Trend Component

The trend component refers to the long-term progression or direction in a time series data, independent of any seasonal effects. It is often represented by a linear or non-linear function added to a model to account for gradual increases or decreases in the data over time, thus capturing sustained movements in the series.

Transcript

00:01 A part is at t equals to 1, which is january 2014, putting the value we get as equals to 2 .7 plus 0 .142, multiply by 1 plus 2 .2, multiply by 2 .2, multiply by sine of pi, multiply by 1 ,000.

00:36 Divided by 6 minus pi by 2 by by solving this we get s equals to 2 .7 plus 0 .142 plus 2 .2 sine of pi by 3 minus of pi by 3 sign is an odd function so that s is equal to 2 .7 plus 0 .142 minus 2 .2.

01:23 Sign of pi by 3.

01:28 Now by putting the value, we get as equals to 2 .7 plus 0 .142 minus 2 .2, multiply by root 3 divided by 2.

01:43 By solving this, we get as equals to 0 .937, which is equal to 937, which is equal to 9 .9.

01:55 94 units.

02:00 Now b parties at t equals to 14 february 2015.

02:16 Putting the value we get as equals to 2 .7 plus 0 .142 multiply by 14 plus 2 .2 multiply by sine of pi multiplied by 14.

02:33 By 6 minus pi by 2 by solving this we get as equals to 2 .7 plus 1 .988 plus 2 .2 sign of 11 pi by 6 the angle lies in quadrant 4th where sign is negative the reference angle is 2 pi minus 11 pi divided by 6 which is equals to minus of pi by 6 as equals to 2 .7 plus 1 .988 plus 2 .m minus 2 .2 sine of pi by 6.

04:01 By solving this we get 2 .7 plus 1 .9 .8 minus 2 .2 multiply by 1 divided by 2.

04:14 And by solving this we get s equals to 3 .5 .8 which means 359 units.

04:27 Now c part is at t equals to 5 .5 .5.

04:37 Which means may 2014...

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