Question

1. is the data linear or exponential?
2. write the equation of the line of best fit as given by the calculator. round to the nearest whole number.
3. write the correlation coefficient. round to the nearest thousandths
4. write a sentence explaining what the slope/

growth factor represents.

1. write a sentence explaining what the y - intercept/initial value represents.
2. predict the number of horses in 2020.

© bench/moulton 2016 - 17
test 5.1 next time, study #5 - 1, #5 - 2, #5 - 3 and #5 - 4

Explanation:

Step1: Confirm linearity (Q17)

Check rate of change consistency:
$\Delta y_1 = 9-1=8, \Delta x_1=5-2=3$, rate $\frac{8}{3}\approx2.67$
$\Delta y_2 = 18-9=9, \Delta x_2=8-5=3$, rate $\frac{9}{3}=3$
$\Delta y_3 = 26-18=8, \Delta x_3=10-8=2$, rate $\frac{8}{2}=4$
$\Delta y_4 = 40-26=14, \Delta x_4=15-10=5$, rate $\frac{14}{5}=2.8$
$\Delta y_5 = 43-40=3, \Delta x_5=16-15=1$, rate $\frac{3}{1}=3$
$\Delta y_6 = 50-43=7, \Delta x_6=20-16=4$, rate $\frac{7}{4}=1.75$
$\Delta y_7 = 60-50=10, \Delta x_7=24-20=4$, rate $\frac{10}{4}=2.5$
Rates are close, so data is linear.

Step2: Calculate line of best fit (Q18)

Use linear regression formula. Let $x$ = years since 1950, $y$ = # of horses.
First, calculate sums:
$n=8$
$\sum x=2+5+8+10+15+16+20+24=100$
$\sum y=1+9+18+26+40+43+50+60=247$
$\sum xy=(2*1)+(5*9)+(8*18)+(10*26)+(15*40)+(16*43)+(20*50)+(24*60)=2+45+144+260+600+688+1000+1440=4179$
$\sum x^2=2^2+5^2+8^2+10^2+15^2+16^2+20^2+24^2=4+25+64+100+225+256+400+576=1650$

Slope $m=\frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$
$m=\frac{8*4179 - 100*247}{8*1650 - 100^2}=\frac{33432-24700}{13200-10000}=\frac{8732}{3200}\approx2.73$
Round to whole number: $m=3$

Y-intercept $b=\frac{\sum y - m\sum x}{n}$
$b=\frac{247 - 3*100}{8}=\frac{247-300}{8}=\frac{-53}{8}\approx-6.63$
Round to whole number: $b=-7$
Equation: $y=3x-7$

Step3: Calculate correlation coefficient (Q19)

Use formula $r=\frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$
First, $\sum y^2=1^2+9^2+18^2+26^2+40^2+43^2+50^2+60^2=1+81+324+676+1600+1849+2500+3600=10631$

Numerator: $8*4179 - 100*247=8732$
Denominator part1: $8*1650 - 100^2=3200$
Denominator part2: $8*10631 - 247^2=85048-61009=24039$
Denominator: $\sqrt{3200*24039}=\sqrt{76924800}\approx8770.68$
$r=\frac{8732}{8770.68}\approx0.9956$
Round to nearest thousandths: $r=0.996$

Step4: Interpret slope (Q20)

Slope is 3, so it represents the approximate annual increase in the number of horses.

Step5: Interpret y-intercept (Q21)

Y-intercept is -7, which represents the estimated number of horses in 1950 (when $x=0$); the negative value is a model approximation since horse count can't be negative.

Step6: Predict 2028 horse count (Q22)

Calculate $x$ for 2028: $2028-1950=78$
Substitute into $y=3x-7$:
$y=3*78 -7=234-7=227$

Answer:

1. Linear
2. $y=3x-7$
3. $0.996$
4. The slope means the number of horses increases by approximately 3 each year.
5. The y-intercept is the estimated (negative, approximate) number of horses in 1950, a model prediction.
6. 227