Question

the data show the chest size and weight of several bears. find the regression equation, letting chest size be the independent (x) variable. then find the best predicted weight of a bear with a chest size of 51 inches. is the result close to the actual weight of 442 pounds? use a significance level of 0.05.
chest size (inches) 45 50 43 43 52 52
weight (pounds) 352 374 275 314 440 367
click the icon to view the critical values of the pearson correlation coefficient r.
what is the regression equation?
ŷ = □ + □x (round to one decimal place as needed.)

Explanation:

Step1: Calculate necessary sums

First, we list the data points:
\(x\) (chest size): 45, 50, 43, 43, 52, 52
\(y\) (weight): 352, 374, 275, 314, 440, 367

Calculate \(n = 6\) (number of data points).

Calculate \(\sum x\): \(45 + 50 + 43 + 43 + 52 + 52 = 285\)

Calculate \(\sum y\): \(352 + 374 + 275 + 314 + 440 + 367 = 2122\)

Calculate \(\sum xy\): \(45\times352 + 50\times374 + 43\times275 + 43\times314 + 52\times440 + 52\times367\)
\(= 15840 + 18700 + 11825 + 13502 + 22880 + 19084 = 101831\)

Calculate \(\sum x^2\): \(45^2 + 50^2 + 43^2 + 43^2 + 52^2 + 52^2\)
\(= 2025 + 2500 + 1849 + 1849 + 2704 + 2704 = 13631\)

Step2: Calculate slope \(b\) and intercept \(a\)

The formula for the slope \(b\) of the regression line is:
\(b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)

Substitute the values:
\(n = 6\), \(\sum xy = 101831\), \(\sum x = 285\), \(\sum y = 2122\), \(\sum x^2 = 13631\)

\(b = \frac{6\times101831 - 285\times2122}{6\times13631 - 285^2}\)

First, calculate numerator: \(6\times101831 = 610986\), \(285\times2122 = 604770\), so numerator \(= 610986 - 604770 = 6216\)

Denominator: \(6\times13631 = 81786\), \(285^2 = 81225\), so denominator \(= 81786 - 81225 = 561\)

Thus, \(b = \frac{6216}{561} \approx 11.1\) (rounded to one decimal place)

The formula for the intercept \(a\) is:
\(a = \bar{y} - b\bar{x}\), where \(\bar{x} = \frac{\sum x}{n}\) and \(\bar{y} = \frac{\sum y}{n}\)

\(\bar{x} = \frac{285}{6} = 47.5\)

\(\bar{y} = \frac{2122}{6} \approx 353.6667\)

\(a = 353.6667 - 11.1\times47.5\)

\(11.1\times47.5 = 527.25\)

\(a = 353.6667 - 527.25 \approx -173.6\) (rounded to one decimal place)

Answer:

\(\hat{y} = -173.6 + 11.1x\)