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QuestionThe concert venue's attendance drops with snow. Analyze data, find the regression line, and predict attendance at 9.1 inches.
(a) Is a linear model suitable? Yes/No (b) Find slope and yy-intercept: Attendance == (Inches of Snow) + (c) Predict attendance with 9.1 inches of snow (nearest integer).

Studdy Solution

STEP 1

Assumptions1. The maximum attendance is7,500 people. The attendance decreases linearly with the increase in snowfall3. The data provided is accurate and sufficient for a linear regression model

STEP 2

We will use the method of least squares to find the best fit line for the given data. The formula for the slope (m) of the best fit line ism=n(∑xy)−(∑x)(∑y)n(∑x2)−(∑x)2m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}where- n is the number of data points- ∑xy\sum xy is the sum of the product of x and y values- ∑x\sum x is the sum of x values- ∑y\sum y is the sum of y values- ∑x2\sum x^2 is the sum of squares of x values

STEP 3

Let's calculate the necessary sums for the formula. We denote "Inches of Snow" as x and "Attendance" as y.∑x=14+0+7+2+6+1+10\sum x =14 +0 +7 +2 +6 +1 +10∑y=0+7500+401+6539+5126+7500+1200\sum y =0 +7500 +401 +6539 +5126 +7500 +1200∑xy=14∗0+0∗7500+7∗401+2∗6539+6∗5126+1∗7500+10∗1200\sum xy =14*0 +0*7500 +7*401 +2*6539 +6*5126 +1*7500 +10*1200∑x2=142+02+72+22+62+12+102\sum x^2 =14^2 +0^2 +7^2 +2^2 +6^2 +1^2 +10^2

STEP 4

Now, plug in the values into the formula for the slope.
m=7(∑xy)−(∑x)(∑y)7(∑x2)−(∑x)2m = \frac{7(\sum xy) - (\sum x)(\sum y)}{7(\sum x^2) - (\sum x)^2}

STEP 5

The formula for the y-intercept (b) of the best fit line isb=(∑y)(∑x2)−(∑x)(∑xy)n(∑x2)−(∑x)2b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}

STEP 6

Now, plug in the values into the formula for the y-intercept.
b=(∑y)(∑x2)−(∑x)(∑xy)(∑x2)−(∑x)2b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{(\sum x^2) - (\sum x)^2}

STEP 7

The equation of the best fit line isAttendance=m∗(Inches of Snow)+bAttendance = m * (Inches\, of\, Snow) + b

STEP 8

To predict the attendance at a concert when there are.1 inches of snow on the ground, plug in the value of.1 for "Inches of Snow" in the equation.
Predicted Attendance=m∗.1+bPredicted\, Attendance = m *.1 + b

STEP 9

Round the predicted attendance to the nearest integer.

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