Question

1. the scatterplot and table show the season number of several television shows and the number of viewers. a linear function can be used to model this relationship. what is the function that best models this relationship? round to the nearest hundredth.

television show viewers
number of seasons, x: 1, 2, 2, 3, 3, 4, 4, 5, 6, 6
number of viewers (millions), y: 53, 49, 44, 45, 42, 44, 34, 31, 32, 28
(scatterplot with x as season number and y as number of viewers (millions))
enter your answer in the box.

Explanation:

Step1: Identify the data points

We have the following data points from the table: \((1, 53)\), \((2, 49)\), \((2, 44)\), \((3, 45)\), \((3, 42)\), \((4, 44)\), \((4, 34)\), \((5, 31)\), \((6, 32)\), \((6, 28)\)

Step2: Calculate the slope (\(m\)) of the linear regression

The formula for the slope of the linear regression line is:

$$ m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} $$

First, we calculate the necessary sums:

• \(n = 10\) (number of data points)
• \(\sum x=1 + 2+2 + 3+3 + 4+4 + 5+6+6=36\)
• \(\sum y=53 + 49+44 + 45+42 + 44+34 + 31+32+28 = 402\)
• \(\sum xy=(1\times53)+(2\times49)+(2\times44)+(3\times45)+(3\times42)+(4\times44)+(4\times34)+(5\times31)+(6\times32)+(6\times28)\)

$$
LATEXBLOCK0
$$

• \(\sum x^{2}=1^{2}+2^{2}+2^{2}+3^{2}+3^{2}+4^{2}+4^{2}+5^{2}+6^{2}+6^{2}\)

$$
LATEXBLOCK1
$$

Now, substitute these values into the slope formula:

$$ m=\frac{10\times1327-36\times402}{10\times147-(36)^{2}} $$

$$
LATEXBLOCK2
$$

Step3: Calculate the y-intercept (\(b\))

The formula for the y-intercept is:

$$ b=\frac{\sum y - m\sum x}{n} $$

Substitute \(m\approx - 6.908\), \(\sum y = 402\), \(\sum x = 36\) and \(n = 10\):

$$ b=\frac{402-(-6.908)\times36}{10}=\frac{402 + 248.688}{10}=\frac{650.688}{10}=65.0688 $$

Step4: Write the linear function

The linear function is of the form \(y = mx + b\). Substituting \(m\approx - 6.91\) (rounded to the nearest hundredth) and \(b\approx65.07\) (rounded to the nearest hundredth), we get:

$$ y=-6.91x + 65.07 $$

Answer:

The linear function that best models the relationship is \(y = - 6.91x+65.07\) (the values of \(m\) and \(b\) are rounded to the nearest hundredth)