Question

2 - 105. tim is buying snacks for the mathletes, who love microwave popcorn. when tim looks at the popcorn selection he notices many brands and different prices. he wonders if the cost is related to the quality of the popcorn. to answer his question, he purchases a random sample of popcorn bags and records their price. when it is time for the mathletes meeting he pops each bag in the same microwave, opens each bag, and counts the number of unpopped kernels. 2 - 105 hw etool homework help

a. make a scatterplot on graph paper and draw the line of best fit. choose two points on your drawn line and use them to determine the equation of the line of best fit. what do the slope and the y - intercept represent in your equation?
b. estimate the number of unpopped kernels (after cooking) in a bag that costs $1.19.

Explanation:

Step1: Assume two points on the line of best fit

Let's assume two points on the line of best fit. For example, assume the points \((x_1,y_1)=(0.6,30)\) and \((x_2,y_2)=(2.3,4)\)

Step2: Calculate the slope \(m\)

The formula for the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(x_1 = 0.6,y_1=30,x_2 = 2.3,y_2 = 4\) into the formula:

$$m=\frac{4 - 30}{2.3-0.6}=\frac{- 26}{1.7}\approx - 15.29$$

Step3: Use the point - slope form \(y - y_1=m(x - x_1)\) to find the equation of the line

Using the point \((x_1,y_1)=(0.6,30)\) and \(m\approx - 15.29\), we have \(y-30=-15.29(x - 0.6)\)

$$y-30=-15.29x+9.174$$

$$y=-15.29x + 39.174$$

The slope \(-15.29\) represents the rate of change of the number of un - popped kernels with respect to the price. For every one - dollar increase in price, the number of un - popped kernels decreases by approximately \(15.29\). The \(y\) - intercept \(39.174\) represents the estimated number of un - popped kernels when the price of the popcorn bag is \(x = 0\) (in a theoretical sense, as a price of \(0\) may not be practical in this context)

Step4: Estimate the number of un - popped kernels for \(x = 1.19\)

Substitute \(x = 1.19\) into the equation \(y=-15.29x + 39.174\)

$$y=-15.29\times1.19+39.174$$

$$y=-18.1951+39.174$$

$$y\approx20.98\approx21$$

Answer:

a. The equation of the line of best fit (using assumed points) is \(y=-15.29x + 39.174\). The slope \(-15.29\) represents the decrease in the number of un - popped kernels per dollar increase in price. The \(y\) - intercept \(39.174\) represents the estimated number of un - popped kernels when the price is \(0\).
b. The estimated number of un - popped kernels for a bag that costs \(\$1.19\) is \(21\)