Question

the ordered pairs show amounts y (in inches) of rainfall equivalent to x inches of snow.(16, 1.5), (12, 1.3), (18, 1.8), (15, 1.5), (20, 2.1), (23, 2.4)about how many inches of rainfall are equivalent to 6 inches of snow? round your answer to the nearest hundredth, if necessary.about $\boldsymbol{square}$ inches of rainjustify your answer.

Explanation:

Step1: Identify linear regression model

We will use linear regression to find the relationship between snow \(x\) and rain \(y\). The linear model is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step2: Calculate necessary sums

First, list the data points \((x_i, y_i)\):
\((16,1.5), (12,1.3), (18,1.8), (15,1.5), (20,2.1), (23,2.4)\)
Calculate sums:
\(\sum x_i = 16+12+18+15+20+23 = 104\)
\(\sum y_i = 1.5+1.3+1.8+1.5+2.1+2.4 = 10.6\)
\(\sum x_i y_i = (16×1.5)+(12×1.3)+(18×1.8)+(15×1.5)+(20×2.1)+(23×2.4) = 24 + 15.6 + 32.4 + 22.5 + 42 + 55.2 = 191.7\)
\(\sum x_i^2 = 16^2+12^2+18^2+15^2+20^2+23^2 = 256+144+324+225+400+529 = 1878\)
\(n = 6\) (number of data points)

Step3: Compute slope \(m\)

Use formula:
$$m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$$
Substitute values:
$$m = \frac{6×191.7 - 104×10.6}{6×1878 - 104^2} = \frac{1150.2 - 1102.4}{11268 - 10816} = \frac{47.8}{452} ≈ 0.10575$$

Step4: Compute intercept \(b\)

Use formula:
$$b = \frac{\sum y_i - m\sum x_i}{n}$$
Substitute values:
$$b = \frac{10.6 - 0.10575×104}{6} = \frac{10.6 - 10.998}{6} = \frac{-0.398}{6} ≈ -0.0663$$

Step5: Predict \(y\) for \(x=6\)

Substitute \(x=6\) into \(y = mx + b\):
$$y = 0.10575×6 + (-0.0663) = 0.6345 - 0.0663 = 0.5682$$
Round to nearest hundredth: \(y ≈ 0.57\)

Answer:

about 0.57 inches of rain

Justification: A linear regression model \(y \approx 0.1058x - 0.0663\) was fitted to the given (snow, rain) data points. Substituting \(x=6\) (inches of snow) into the model gives the predicted rainfall value, rounded to 0.57 inches.