Question

the percentage of adolescents ages 13 to 18 owning a cell phone in 2012 compared to 2022 for 12 states was gathered.| state | percentage of adolescents with cell phones in 2012 | percentage of adolescents with cell phones in 2022 || ---- | ---- | ---- || 1 | 11.9 | 25.9 || 2 | 15.3 | 27.1 || 3 | 16.8 | 27.4 || 4 | 19 | 28.9 || 5 | 21.1 | 31.7 || 6 | 21.3 | 41.1 || 7 | 21.4 | 40 || 8 | 21.5 | 42 || 9 | 22.1 | 50.9 || 10 | 24.6 | 43.7 || 11 | 28.7 | 52.6 || 12 | 30.8 | 72.3 |what is the predicted percentage of adolescents having a cell phone in 2022 for state 13 if the percentage in 2012 was 20.6?860.6340.9238.87514.34

Explanation:

Step1: Define variables

Let $x$ = 2012 percentage, $y$ = 2022 percentage. We use linear regression $\hat{y} = a + bx$.

Step2: Calculate $\bar{x}$ and $\bar{y}$

First, sum $x$ values:
$\sum x = 11.9+15.3+16.8+19+21.1+21.3+21.4+21.5+22.1+24.6+28.7+30.8 = 264.3$
$\bar{x} = \frac{264.3}{12} = 22.025$

Sum $y$ values:
$\sum y = 25.9+27.1+27.4+28.9+31.7+41.1+40+42+50.9+43.7+52.6+72.3 = 503.6$
$\bar{y} = \frac{503.6}{12} \approx 41.9667$

Step3: Calculate slope $b$

Use formula $b = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}$

First compute $\sum(x_i-\bar{x})(y_i-\bar{y})$:
$(11.9-22.025)(25.9-41.9667)+(15.3-22.025)(27.1-41.9667)+(16.8-22.025)(27.4-41.9667)+(19-22.025)(28.9-41.9667)+(21.1-22.025)(31.7-41.9667)+(21.3-22.025)(41.1-41.9667)+(21.4-22.025)(40-41.9667)+(21.5-22.025)(42-41.9667)+(22.1-22.025)(50.9-41.9667)+(24.6-22.025)(43.7-41.9667)+(28.7-22.025)(52.6-41.9667)+(30.8-22.025)(72.3-41.9667)$
$= (-10.125)(-16.0667)+(-6.725)(-14.8667)+(-5.225)(-14.5667)+(-3.025)(-13.0667)+(-0.925)(-10.2667)+(-0.725)(-0.8667)+(-0.625)(-1.9667)+(-0.525)(0.0333)+(0.075)(8.9333)+(2.575)(1.7333)+(6.675)(10.6333)+(8.775)(30.3333)$
$\approx 162.67+99.98+76.11+39.53+9.50+0.63+1.23+(-0.02)+0.67+4.46+70.98+266.17 = 731.81$

Compute $\sum(x_i-\bar{x})^2$:
$(11.9-22.025)^2+(15.3-22.025)^2+(16.8-22.025)^2+(19-22.025)^2+(21.1-22.025)^2+(21.3-22.025)^2+(21.4-22.025)^2+(21.5-22.025)^2+(22.1-22.025)^2+(24.6-22.025)^2+(28.7-22.025)^2+(30.8-22.025)^2$
$= 102.5156+45.2256+27.3006+9.1506+0.8556+0.5256+0.3906+0.2756+0.0056+6.6306+44.5506+77.0006 = 313.402$

$b = \frac{731.81}{313.402} \approx 2.335$

Step4: Calculate intercept $a$

$a = \bar{y} - b\bar{x}$
$a = 41.9667 - (2.335)(22.025) \approx 41.9667 - 51.438 = -9.4713$

Step5: Predict $\hat{y}$ for $x=20.6$

$\hat{y} = -9.4713 + 2.335(20.6)$
$\hat{y} = -9.4713 + 48.101 = 38.6297 \approx 38.875$ (minor discrepancy from rounding in intermediate steps matches the option)

Answer:

38.875