Example 1 states explicitly to use a sinusoidal regression to model the data. Even if this information is not given, a sinusoidal pattern is easy to recognize if sufficient data points are available.

Recall from Lesson 8A that many natural patterns are sinusoidal. Examples 2 and 3 demonstrate how a sinusoidal function that models naturally occurring data can be used to solve problems.

The table below provides the mean minimum monthly temperature during two years at Winnipeg, Manitoba. Find the sinusoidal regression function for the data and use it to find the mean minimum monthly temperature for January, 2009.

2007
T min (°C)
2008
T min (°C)
1
Jan.
–24.5
13
Jan.
–18.4
2
Feb.
–20.0
14
Feb.
–7.6
3
Mar.
–15.8
15
Mar.
–9.7
4
Apr.
–5.7
16
Apr.
0.2
5
May
0.7
17
May
4.7
6
June
11.4
18
June
10.7
7
July
13.9
19
July
12.4
8
Aug.
11.1
20
Aug.
13.3
9
Sept.
7.1
21
Sept.
6.9
10
Oct.
0.1
22
Oct.
1.5
11
Nov.
–9.0
23
Nov.
–7.1
12
Dec.
–9.0
24
Dec.
–17.4

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Note: To find a regression equation only, omit steps 1, 2, 3, and 7.

Step 2: Clear the functions; press Y= then, CLEAR for each function that must be deleted.

Step 3: Select values for the viewing window; press WINDOW. Use the keypad to type the values from the table. Use the up and down arrow keys to scroll through the list.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted; then, CLEAR, ENTER. Repeat Step 5 for all columns that have unneeded data. Use the left and right arrow keys to move between columns.

Step 6: Enter the data.

Step 7: View the data; press GRAPH.

In Step 10, an estimate of the period is used to improve the accuracy of the sinusoidal regression equation. You will not be required to do this on your assigned questions in this unit.

Step 11: View the scatter plot and regression model; press GRAPH.

The sinusoidal regression equation is y = 15.6479 sin(0.5327x – 2.3357) – 1.8425.




To find the mean minimum monthly temperature for January, 2009, means to find y when x = 25.

Step 1: Go to the CALCULATE menu; press 2nd, TRACE.

Step 2: Select value; press 1.

Step 3: Type the required x-value; press 25.

Step 4: Get the corresponding y-value; press ENTER.

The mean minimum monthly temperature for January 2009 was approximately –17.49°C.