Chapter 23, Problem 23.34E

(a)

To determine

To find:

Theequation of the least-squares line of the given data.

Answer to Problem 23.34E

  $Y=\mathrm{31.93430.304}X$

Explanation of Solution

Given:

The previous year’s adult birds return $=60\%$

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

The independent variable is $X$

The dependent variable is $Y$ .

The regression coefficients,

Draw the table

$X$	$Y$	$X\ast Y$	$X^2$	$Y^2$
$\text{74}$	$\text{5}$	$\text{370}$	$\text{5476}$	$\text{25}$
$\text{66}$	$\text{6}$	$\text{396}$	$\text{4356}$	$\text{36}$
$\text{81}$	$\text{8}$	$\text{648}$	$\text{6561}$	$\text{64}$
$\text{52}$	$\text{11}$	$\text{572}$	$\text{2704}$	$\text{121}$
$\text{73}$	$\text{12}$	$\text{876}$	$\text{5329}$	$\text{144}$
$\text{62}$	$\text{15}$	$\text{930}$	$\text{3844}$	$\text{225}$
$\text{52}$	$\text{16}$	$\text{832}$	$\text{2704}$	$\text{256}$
$\text{45}$	$\text{17}$	$\text{765}$	$\text{2025}$	$\text{289}$
$\text{62}$	$\text{18}$	$\text{1116}$	$\text{3844}$	$\text{324}$
$\text{46}$	$\text{18}$	$\text{828}$	$\text{2116}$	$\text{324}$
$\text{60}$	$19$	$\text{1140}$	$\text{3600}$	$\text{361}$
$\text{46}$	$\text{20}$	$\text{920}$	$\text{2116}$	$\text{400}$
$\text{38}$	$\text{20}$	$\text{760}$	$\text{1444}$	$\text{400}$
${\displaystyle \sum X}=757$	${\displaystyle \sum Y}=185$	${\displaystyle \sum XY}=10153$	${\displaystyle \sum X^2}=46119$	${\displaystyle \sum Y^2}=2969$

  $\begin{array}{l}757X_i=X=ny58.230769230769\\ \Sigma Y=185=14.23076923076913\\ ssxx=\Sigma x^{2-1}\Sigma x^i)46119-7572/13\\ ssxx=2038.3076923077\\ ss\\ Y=\Sigma Y\end{array}$

Therefore, the regression coefficients (the slope m, and the y- intercept n).

The regression equation is

  $Y=\mathrm{31.93430.304}X$

(b)

To determine

To find:

The equation of the least-squares line of the given data.

Answer to Problem 23.34E

  $Y-31.9343-0.304*6013.6943$

Explanation of Solution

Given:

The previous year’s adult birds return $=60\%$

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

The independent variable is $X$

The dependent variable is $Y$ .

The regression coefficients,

Draw the table

$X$	$Y$	$X\ast Y$	$X^2$	$Y^2$
$\text{74}$	$\text{5}$	$\text{370}$	$\text{5476}$	$\text{25}$
$\text{66}$	$\text{6}$	$\text{396}$	$\text{4356}$	$\text{36}$
$\text{81}$	$\text{8}$	$\text{648}$	$\text{6561}$	$\text{64}$
$\text{52}$	$\text{11}$	$\text{572}$	$\text{2704}$	$\text{121}$
$\text{73}$	$\text{12}$	$\text{876}$	$\text{5329}$	$\text{144}$
$\text{62}$	$\text{15}$	$\text{930}$	$\text{3844}$	$\text{225}$
$\text{52}$	$\text{16}$	$\text{832}$	$\text{2704}$	$\text{256}$
$\text{45}$	$\text{17}$	$\text{765}$	$\text{2025}$	$\text{289}$
$\text{62}$	$\text{18}$	$\text{1116}$	$\text{3844}$	$\text{324}$
$\text{46}$	$\text{18}$	$\text{828}$	$\text{2116}$	$\text{324}$
$\text{60}$	$19$	$\text{1140}$	$\text{3600}$	$\text{361}$
$\text{46}$	$\text{20}$	$\text{920}$	$\text{2116}$	$\text{400}$
$\text{38}$	$\text{20}$	$\text{760}$	$\text{1444}$	$\text{400}$
${\displaystyle \sum X}=757$	${\displaystyle \sum Y}=185$	${\displaystyle \sum XY}=10153$	${\displaystyle \sum X^2}=46119$	${\displaystyle \sum Y^2}=2969$

  $\begin{array}{l}757X_i=X=ny58.230769230769\\ \Sigma Y=185=14.23076923076913\\ ssxx=\Sigma x^{2-1}\Sigma x^i)46119-7572/13\\ ssxx=2038.3076923077\\ ss\\ Y=\Sigma Y\end{array}$

Therefore, the regression coefficients (the slope m, and the y- intercept n).

The regression equation is

  $Y=\mathrm{31.93430.304}X$

The slope of the regression line is

  $m=-0.304$

Since the sign is negative,

Therefore the slope is a negative linear relationship between the percent return and new adults.

Also for every $1$ percent increase of adult birds in a colony that return from the previous year, number of new adults that join the colony decreases by a factor of $0.304$

By using $X=60$

  $Y-31.9343-0.304*6013.6943$