Question

listed below are the overhead widths (cm) of seals measured from photographs and weights (kg) of the seals. find the 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 2.3 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) | 7.2 | 7.4 | 9.8 | 9.3 | 8.7 | 8.4
weight (kg) | 121 | 159 | 252 | 205 | 203 | 195

the regression equation is ŷ = □ + (□)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 \( x^2 \).
Given \( x \) values: \( 7.2, 7.4, 9.8, 9.3, 8.7, 8.4 \)
Given \( y \) values: \( 121, 159, 252, 205, 203, 196 \)

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

\( \sum x = 7.2 + 7.4 + 9.8 + 9.3 + 8.7 + 8.4 = 50.8 \)

\( \sum y = 121 + 159 + 252 + 205 + 203 + 196 = 1136 \)

\( \sum xy = (7.2 \times 121) + (7.4 \times 159) + (9.8 \times 252) + (9.3 \times 205) + (8.7 \times 203) + (8.4 \times 196) \)
\( = 871.2 + 1176.6 + 2469.6 + 1906.5 + 1766.1 + 1646.4 = 9836.4 \)

\( \sum x^2 = (7.2^2) + (7.4^2) + (9.8^2) + (9.3^2) + (8.7^2) + (8.4^2) \)
\( = 51.84 + 54.76 + 96.04 + 86.49 + 75.69 + 70.56 = 435.38 \)

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

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

Substitute the values:
\( b = \frac{6 \times 9836.4 - 50.8 \times 1136}{6 \times 435.38 - (50.8)^2} \)

First, calculate numerator:
\( 6 \times 9836.4 = 59018.4 \)
\( 50.8 \times 1136 = 50.8 \times 1136 = 57708.8 \)
Numerator: \( 59018.4 - 57708.8 = 1309.6 \)

Denominator:
\( 6 \times 435.38 = 2612.28 \)
\( (50.8)^2 = 2580.64 \)
Denominator: \( 2612.28 - 2580.64 = 31.64 \)

So, \( b = \frac{1309.6}{31.64} \approx 41.4 \) (rounded to one decimal place)

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

Substitute the values:
\( a = \frac{1136 - 41.4 \times 50.8}{6} \)

Calculate \( 41.4 \times 50.8 = 41.4 \times 50 + 41.4 \times 0.8 = 2070 + 33.12 = 2103.12 \)

\( 1136 - 2103.12 = -967.12 \)

\( a = \frac{-967.12}{6} \approx -161 \) (rounded to the nearest integer)

Step3: Write the regression equation

The regression equation is \( \hat{y} = a + bx \), so substituting \( a \) and \( b \):
\( \hat{y} = -161 + 41.4x \)

Answer:

The regression equation is \( \hat{y} = \boxed{-161} + \boxed{41.4}x \).