Question

listed below are the overhead widths (cm) of seals measured from photographs and weights (kg) of the seals. find th regression equation, letting the overhead width be the predictor (x) variable. find the best predicted weight of a seal if the overhead width measured from a photograph is 1.7 cm, using the regression equation. can the prediction be correct? if not, what is wrong? use a significance level of 0.05.
overhead width (cm) | 7.1 | 7.4 | 9.8 | 9.4 | 8.9 | 8.5
weight (kg) | 133 | 176 | 272 | 228 | 229 | 218

the regression equation is \\(\hat{y} = \square + (\square)x\\).
(round the y-intercept to the nearest integer as needed. round the slope to one decimal place as needed.)

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums of \(x\), \(y\), \(xy\), and \(x^2\). Let \(x\) be the overhead width and \(y\) be the weight.

Given data:
\(x\): 7.1, 7.4, 9.8, 9.4, 8.9, 8.5
\(y\): 133, 176, 272, 228, 229, 218

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

Sum of \(x\) (\(\sum x\)): \(7.1 + 7.4 + 9.8 + 9.4 + 8.9 + 8.5 = 51.1\)

Sum of \(y\) (\(\sum y\)): \(133 + 176 + 272 + 228 + 229 + 218 = 1256\)

Sum of \(xy\) (\(\sum xy\)):
\((7.1\times133)+(7.4\times176)+(9.8\times272)+(9.4\times228)+(8.9\times229)+(8.5\times218)\)
\(= 944.3 + 1294.4 + 2665.6 + 2143.2 + 2038.1 + 1853 = 10938.6\)

Sum of \(x^2\) (\(\sum x^2\)):
\(7.1^2 + 7.4^2 + 9.8^2 + 9.4^2 + 8.9^2 + 8.5^2\)
\(= 50.41 + 54.76 + 96.04 + 88.36 + 79.21 + 72.25 = 440.03\)

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

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

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

First, calculate \(\bar{x}\) and \(\bar{y}\):
\(\bar{x}=\frac{51.1}{6}\approx8.5167\)
\(\bar{y}=\frac{1256}{6}\approx209.3333\)

Now calculate \(b\):

$$
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$$

Now calculate \(a\):
\(a = \bar{y}-b\bar{x}\)
\(a = 209.3333 - 49.9\times8.5167\)
\(a = 209.3333 - 425.0833\)
\(a\approx - 215.75\) (Rounded to nearest integer: -216)

So the regression equation is \(\hat{y}=a + bx=-216 + 49.9x\)

Answer:

The regression equation is \(\hat{y}=\boxed{-216} + \boxed{49.9}x\)