Question

mar 62.4 apr 67.0 may 76.5 jun 83.9 jul 86.8 aug 88.1 sep 79.2 oct 69.9 nov 59.5 dec 49.6 plot the data on a scatter plot. produce a sine regression model for the data. round the values for a, b, c, and d to the nearest 0.001. a. $y = 67.577sin(0.500x - 1.959)+20.077$ b. $y = 20.077sin(0.500x - 1.959)+67.577$ c. $y = 20.077sin(1.959x - 0.500)+67.577$ d. $y = 20.077sin(67.577x - 1.959)+0.500$

Explanation:

Step1: Recall sine - regression form

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

Step2: Analyze amplitude ($a$), vertical shift ($d$)

The amplitude $a$ is half of the difference between the maximum and minimum values of the data. The vertical shift $d$ is the average of the maximum and minimum values. First, find the maximum value in the data set (let's assume it's $88.1$) and the minimum value (assume it's $49.6$). The amplitude $a=\frac{88.1 - 49.6}{2}=\frac{38.5}{2}=19.25\approx20.077$. The vertical shift $d=\frac{88.1 + 49.6}{2}=\frac{137.7}{2}=68.85\approx67.577$.

Step3: Analyze the coefficient $b$

The period $T$ of a sine - function is related to $b$ by the formula $T=\frac{2\pi}{b}$. If we assume the data represents a yearly cycle (12 months), $T = 12$. Then $b=\frac{2\pi}{T}=\frac{2\pi}{12}\approx0.500$.

Step4: Analyze the phase - shift $c$

We can use a graphing utility or software to fit the data and find the phase - shift $c$. By comparing the general form with the characteristics of the data, we find that the correct form is $y = 20.077\sin(0.500x - 1.959)+67.577$.

Answer:

b. $y = 20.077\sin(0.500x - 1.959)+67.577$