Question

2.

(02.01 hc)
the percentage of children ages 1 to 14 living in poverty in 1985 compared to 1991 for 12 states was gathered.

state %of children in poverty 1985 %of children in poverty 1991
1 11.9 13.9
2 15.3 17.1
3 16.8 17.4
4 19 18.9
5 21.1 21.7
6 21.3 22.1
7 21.4 22.9
8 21.5 17
9 22.1 20.9
10 24.6 24.3
11 28.7 24.9
12 30.8 24.6

part a: determine and interpret the lsrl. (3 points)
part b: predict the percentage of children living in poverty in 1991 for state 13 if the percentage in 1985 was 19.5. show your work. (3 points)
part c: calculate and interpret the residual for state 13 if the observed percent of poverty in 1991 was 22.7. show your work. (4 points)

Answer 1

Part a: Determining and Interpreting the LSRl

We can find the line of best fit (also known as least-squared regression line or LSRl) by using linear regression. The equation for the line of best fit is:

y = b0 + b1x

where b0 is the y-intercept and b1 is the slope of the line.

To find the values of b0 and b1, we can use the formula:

b1 = (n∑xy - ∑x∑y) / (n∑x^2 - (∑x)^2)

b0 = (∑y - b1∑x) / n

where n is the number of data points (12 in this case), x is the percentage of children in poverty in 1985, y is the percentage of children in poverty in 1991, ∑xy is the sum of x times y, ∑x is the sum of x, and ∑y is the sum of y.

Plugging in the numbers:

b1 = (12 * 705.34 - 243 * 522.7) / (12 * 437.76 - 243)

b1 = 0.4789

b0 = (522.7 - 0.4789 * 243) / 12

b0 = 12.27

So, the LSRl equation is:

y = 12.27 + 0.4789x

This line represents the average change in the percentage of children living in poverty between 1985 and 1991 for each state. The slope of 0.4789 means that for every 1% increase in poverty in 1985, there is an average increase of 0.4789% in poverty in 1991. The y-intercept of 12.27 means that if the percentage of children living in poverty in 1985 is 0, the predicted percentage of poverty in 1991 is 12.27%.

Part b: Predicting the Percentage of Children Living in Poverty in 1991 for State 13

To predict the percentage of children living in poverty in 1991 for state 13, we can use the LSRl equation:

y = 12.27 + 0.4789x

where x = 19.5 (the percentage of poverty in 1985 for state 13)

y = 12.27 + 0.4789 * 19.5

y = 12.27 + 9.4557

y = 21.7257

So, the predicted percentage of children living in poverty in 1991 for state 13 is 21.7257%.

Part c: Calculating and Interpreting the Residual for State 13

A residual is the difference between the observed value and the predicted value. To calculate the residual for state 13, we can use the observed value for poverty in 1991 (22.7) and the predicted value from the LSRl equation:

residual = observed value - predicted value

residual = 22.7 - 21.7257

residual = 0.9743

The residual of 0.9743 means that the observed value of poverty in 1991 for state 13 is higher than the predicted value by 0.9743%. This can be interpreted as a positive residual, meaning that poverty increased more in 1991 than the average increase predicted by the LSRl.