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4. Use sinusoidal regression to determine the temperatures for Lethbridge in April and December from the following table. (4 marks)

| Month | Average High (°C) |
|-----------|--------------------|
| January | -1.8 |
| February | 1.5 |
| March | 6.0 |
| April | ? ? |
| May | 18.2 |
| June | 22.3 |
| July | 25.5 |
| August | 25.4 |
| September | 20.1 |
| October | 14 |
| November | 4.3 |
| December | ? ? |

Sagot :

GinnyAnswer
To determine the temperatures for Lethbridge in April and December using sinusoidal regression, we follow these steps:

### Step 1: Understand the Given Data

We have data for the average high temperatures in degrees Celsius for most months of the year, except for April and December:

1. January: -1.8°C
2. February: 1.5°C
3. March: 6.0°C
4. April: ? ?
5. May: 18.2°C
6. June: 22.3°C
7. July: 25.5°C
8. August: 25.4°C
9. September: 20.1°C
10. October: 14.0°C
11. November: 4.3°C
12. December: ? ?

### Step 2: Identify the Model for Sinusoidal Regression

The sinusoidal function typically used for modeling seasonal temperature variations is given by:
[tex]\[ T(x) = a \sin(bx + c) + d \][/tex]

Where:
- [tex]\( T(x) \)[/tex] is the temperature at month [tex]\( x \)[/tex].
- [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are parameters of the sinusoidal function to be determined.

### Step 3: Obtain the Parameters from Regression

Through sinusoidal regression, we have the determined parameters:
[tex]\[ a = -13.540836246213875 \][/tex]
[tex]\[ b = 0.5127422277416208 \][/tex]
[tex]\[ c = 1.0814742730215006 \][/tex]
[tex]\[ d = 12.102966612169684 \][/tex]

### Step 4: Calculate the Temperature for April

April corresponds to month 4. Substitute [tex]\( x = 4 \)[/tex] into the sinusoidal equation:
[tex]\[ T(4) = -13.540836246213875 \sin(0.5127422277416208 \times 4 + 1.0814742730215006) + 12.102966612169684 \][/tex]

Computing this:
[tex]\[ T(4) \approx 11.979076871094815 \][/tex]

Thus, the estimated average high temperature for April is:
[tex]\[ 11.98^\circ C \][/tex]

### Step 5: Calculate the Temperature for December

December corresponds to month 12. Substitute [tex]\( x = 12 \)[/tex] into the sinusoidal equation:
[tex]\[ T(12) = -13.540836246213875 \sin(0.5127422277416208 \times 12 + 1.0814742730215006) + 12.102966612169684 \][/tex]

Computing this:
[tex]\[ T(12) \approx 1.0792304456988724 \][/tex]

Thus, the estimated average high temperature for December is:
[tex]\[ 1.08^\circ C \][/tex]

### Conclusion

After conducting the sinusoidal regression analysis and performing the necessary calculations, the estimated average high temperatures for Lethbridge are:

- April: [tex]\(11.98^\circ C\)[/tex]
- December: [tex]\(1.08^\circ C\)[/tex]

These results reflect the expected seasonal temperature variations based on the sinusoidal model.
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