Question

1. dr. teodora collects data on the heights and weights of men in a study she is conducting. the data is below.

height (inches)	66	70	78	74	64	67	72	71	69	67
weight (pounds)	120	160	220	165	125	165	180	185	175	145

q 1 find a regression model, w(h), that shows the estimated weight of men based on their height, h. round all numbers to the nearest tenth.

Explanation:

Step1: Recall linear - regression formula

The linear - regression equation is of the form $y = ax + b$, where $y$ is the dependent variable (weight $w$), $x$ is the independent variable (height $h$), $a$ is the slope, and $b$ is the y - intercept. The formulas for $a$ and $b$ are:

$$a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$$

$$b=\bar{y}-a\bar{x}$$

where $n$ is the number of data points, $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$ and $\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i$.

Let $x$ be the height and $y$ be the weight. Here, $n = 10$.

Step2: Calculate $\sum_{i = 1}^{n}x_i$, $\sum_{i = 1}^{n}y_i$, $\sum_{i = 1}^{n}x_i^{2}$, $\sum_{i = 1}^{n}x_iy_i$

\(\sum_{i = 1}^{10}x_i=66 + 70+78+74+64+67+72+71+69+67 = 708\)
\(\sum_{i = 1}^{10}y_i=120 + 160+220+165+125+165+180+185+175+145 = 1640\)
\(\sum_{i = 1}^{10}x_i^{2}=66^{2}+70^{2}+78^{2}+74^{2}+64^{2}+67^{2}+72^{2}+71^{2}+69^{2}+67^{2}\)

$$=4356+4900+6084+5476+4096+4489+5184+5041+4761+4489 = 48886$$

\(\sum_{i = 1}^{10}x_iy_i=66\times120 + 70\times160+78\times220+74\times165+64\times125+67\times165+72\times180+71\times185+69\times175+67\times145\)

$$=7920+11200+17160+12210+8000+10955+12960+13135+12075+9715 = 115335$$

Step3: Calculate $\bar{x}$ and $\bar{y}$

\(\bar{x}=\frac{\sum_{i = 1}^{10}x_i}{10}=\frac{708}{10}=70.8\)
\(\bar{y}=\frac{\sum_{i = 1}^{10}y_i}{10}=\frac{1640}{10}=164\)

Step4: Calculate the slope $a$

$$a=\frac{10\times115335 - 708\times1640}{10\times48886-708^{2}}$$

$$=\frac{1153350-1161120}{488860 - 501264}=\frac{-7770}{-12404}\approx0.6$$

Step5: Calculate the y - intercept $b$

$$b = 164-0.6\times70.8=164 - 42.5=121.5$$

Answer:

$w(h)=0.6h + 121.5$