Predict the number of polio cases

Question

1988	350
1992	138
1996	33
2000	4
2005	3.2
2007	1.3

Year ^ Number of polio cases (thousands)

Predict the number of polio cases in 2014

Answer 1

Bottom Line Up Front: This is a best-fit line problem that is asking us to A) determine a good estimate for a function that relates to variables and B) Calculate the value of one of the variables given a value for the other variable. Following this structure gives us a negative value, see work below, which would indicate either "we expect zero cases of polio in 2026" or "-254.08 cases" depending on how strictly realistic the course is.

We structure our approach to this problem by making the following assumptions

1. That we can model the number of polio cases per year with a linear function of the form y=mx+b

We define x, our independent variable, to be the number of years since 1988 (we use this rather than the actual year for simpler math.

We define y, our dependent variable as the number of polio cases in thousands.

By this definition we can convert the table given into a set of 6 points, (0,350)(4,138)(8,33)(12,4)(17,3.2)(19,1.3).

We then follow the "least square method" to find the line of best fit. The least squares method uses the following steps

1. Calculate the average of each variable in our set of points (x_a and y_a)
2. x_a=(0+4+8+12+17+19)/6=10
3. y_a=(350+138+33+4+3.2+1.3)/6=88.25
4. Calculate the difference between each point and the average point
5. (0-10,350-88.25)(4-10,138-88.25)(8-10,33-88.25)(12-10,4-88.25)(17-10,3.2-88.25)(19-10,1.3-88.25)=(-10,261.75)(-6,49.75)(-2,-55.25)(2,-84.25)(7,-85.05)(9,-86.95)
6. Calculate the slope using the formula m=(∑(x-x_a)(y-y_a)/∑(x-x_a)²
7. m=(((-10)*(261.75))+((-6)*(49.75))+((-2)*(-55.25))+((2)*(-84.25))+((7)*(-85.05))+((9)*(-86.95)))/((-10)²+(-6)²+(-2)²+(2)²+(7)²+(9)²)=((-2617.5)+(-298.5)+(110.5)+(-168.5)+(-595.35)+(-782.55)/(100+36+4+4+49+81)=(-4351.9)/(274)=-15.88
8. Calculate the intercept using the formula b=y_a-mx_a
9. b=(88.25)-(-15.88)(10)=88.25+158.8=247.05

We now have the best fit line y=-15.88x+247.05. We can then plug in the date requested 2014, or x=26 to get our final answer. y=(-15.88)(26)+(158.8)=-254.08 cases. However, this answer does not strictly speaking make sense because it is not possible to have negative numbers of cases. If this were an SAT questions we would expect to see zero as a possible answer, as that is the minimum value of cases it is possible to have. If a course with less strict realism we could also treat -254.08 as a valid answer.