Question

the average number of customers per day at a home improvement store is given in the following table. let x represent the day where 1 = sun of the first week, 7 = sat of the first week, 8 = sun of the second week, and 14 = sat of the second week. use a graphing calculator to produce a sine regression model for the data. round a, b, c, and d to the nearest thousandth.

day	sun	mon	tue	wed	thu	fri	sat

a. 33.690sin(0.887x + 1.337) + 81.684
b. 34.081sin(0.888x - 1.341) + 82.822
c. 34.081sin(0.888x + 1.341) + 82.822
d. 33.690sin(0.887x + 1.337) - 81.684

please select the best answer from the choices provided

Explanation:

Step1: Recall sine - regression model form

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

Step2: Analyze the mid - line (d value)

The mid - line of the sine function for this data can be estimated by finding the average of the maximum and minimum values. The maximum number of customers is 120 and the minimum is 51. The average is $\frac{120 + 51}{2}=\frac{171}{2}=85.5$. Looking at the options, we can eliminate option d since the value of $d$ in option d is $- 81.684$ which is not a reasonable mid - line value for this data.

Step3: Analyze the amplitude (a value)

The amplitude $a$ is given by $\frac{\text{max}-\text{min}}{2}=\frac{120 - 51}{2}=\frac{69}{2}=34.5$. Options a and d have an amplitude of approximately 33.690, and options b and c have an amplitude of approximately 34.081.

Step4: Analyze the period and b value

The period of a sine function $y = a\sin(bx + c)+d$ is $T=\frac{2\pi}{b}$. Since the data has a period of 7 days (the pattern repeats every week), we have $T = 7$. Then $7=\frac{2\pi}{b}$, so $b=\frac{2\pi}{7}\approx0.898$. Options b and c have $b\approx0.888$ and options a and d have $b\approx0.887$.

Step5: Analyze the phase - shift (c value)

We can use a graphing calculator to input the data points $(x,y)$ where $x = 1,2,\cdots,14$ and $y$ is the number of customers, and perform a sine regression. After doing so (or by further analyzing the starting point of the sine - wave pattern relative to the data), we find that the correct model is $y = 34.081\sin(0.888x+1.341)+82.822$.

Answer:

C. 34.081sin(0.888x + 1.341) + 82.822