5. the table below gives the height (in inches) and shoe sizes of six randomly selected men.

if a man has a shoe size of 10.5, what would be his predicted height? (be careful, if height is x,
you are replacing y!)

Question

5. the table below gives the height (in inches) and shoe sizes of six randomly selected men.

if a man has a shoe size of 10.5, what would be his predicted height? (be careful, if height is x,
you are replacing y!)

Accepted Answer

# Explanation:
## Step1: Calculate the regression equation
Let height be $x$ and shoe size be $y$.
First, calculate the means:
$\bar{x}=\frac{67 + 70+73.5+75+78+66}{6}=\frac{429.5}{6}\approx71.58$
$\bar{y}=\frac{8.5 + 9.5+11+12+13+8}{6}=\frac{62}{6}\approx10.33$

Calculate the slope $b$:
$$
\begin{align*}
b&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\
\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(67 - 71.58)(8.5-10.33)+(70 - 71.58)(9.5 - 10.33)+(73.5-71.58)(11 - 10.33)+(75-71.58)(12 - 10.33)+(78-71.58)(13 - 10.33)+(66-71.58)(8 - 10.33)\
&=(-4.58)(-1.83)+(-1.58)(-0.83)+(1.92)(0.67)+(3.42)(1.67)+(6.42)(2.67)+(-5.58)(-2.33)\
&=8.38+1.31+1.29+5.71+17.14+12.90\
&=46.73
\end{align*}
$$
$$
\begin{align*}
\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}&=(67 - 71.58)^{2}+(70 - 71.58)^{2}+(73.5-71.58)^{2}+(75-71.58)^{2}+(78-71.58)^{2}+(66-71.58)^{2}\
&=(-4.58)^{2}+(-1.58)^{2}+(1.92)^{2}+(3.42)^{2}+(6.42)^{2}+(-5.58)^{2}\
&=20.98+2.50+3.69+11.69+41.22+31.14\
&=111.22
\end{align*}
$$
$b=\frac{46.73}{111.22}\approx0.42$

The intercept $a=\bar{y}-b\bar{x}=10.33-0.42\times71.58=10.33 - 30.06=-19.73$

The regression equation is $y = 0.42x-19.73$

## Step2: Solve for $x$ when $y = 10.5$
Substitute $y = 10.5$ into the equation:
$10.5=0.42x-19.73$
$0.42x=10.5 + 19.73$
$0.42x=30.23$
$x=\frac{30.23}{0.42}\approx72$

# Answer:
$72$ inches

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Calculate the regression equation

Step2: Solve for \(x\) when \(y = 10.5\)

Answer: