Answered

Connect with knowledgeable individuals and get your questions answered on IDNLearn.com. Join our knowledgeable community and access a wealth of reliable answers to your most pressing questions.

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 :

GinnyAnsweridn
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.