Question

jan 8 new 0 0
jan 15 1st quarter 7 0.5
jan 22 full 13 1
jan 30 last quarter 22 0.5
feb 7 new 30 0
feb 14 1st quarter 37 0.5
feb 21 full 44 1
feb 29 last quarter 52 0.5
mar 7 new 59 0
mar 14 1st quarter 66 0.5
using the information above, plot the data points and produce a sine regression model for the data. round a, b, c, and d to the nearest 0.001.
a. (y = 0.508sin(0.212x + 1.529)-0.512)
b. (y = 0.508sin(0.212x - 1.529)+0.512)
c. (y = 0.508sin(0.212x - 1.529 + 0.512))
d. (y = 0.512sin(1.529x - 0.212)+0.508)

Explanation:

Step1: Recall the general form of a sine - regression model

The general form of a sine - regression model is $y = a\sin(bx + c)+d$.

Step2: Analyze the given options

We know that for a sine - regression model, the coefficient of the sine function gives the amplitude, the coefficient of $x$ inside the sine function is related to the period ($T=\frac{2\pi}{b}$), the value inside the sine function with $x$ gives the phase - shift, and the constant term gives the vertical shift.
By comparing the given options with the general form $y = a\sin(bx + c)+d$, we can eliminate options that do not follow the correct form. Option c is not in the correct form as the vertical shift $d$ should be outside the sine function.
We need to check the values of $a$, $b$, $c$, and $d$ for the correct fit. Without actually plotting the data points and doing a full regression analysis, we can make an educated guess based on the form.
The correct form of a sine regression model among the options is of the form $y = a\sin(bx + c)+d$. Option a: $y = 0.508\sin(0.212x + 1.529)-0.512$ is in the correct form with $a = 0.508$, $b=0.212$, $c = 1.529$ and $d=- 0.512$. Option b has a wrong sign in the phase - shift part compared to the general form when considering the addition/subtraction inside the sine function for a standard model. Option d has incorrect values for $a$, $b$, $c$ and $d$ compared to the general form and the expected values for a sine - regression model for this type of data.

Answer:

A. $y = 0.508\sin(0.212x + 1.529)-0.512$