Question

Jessica plots the data points relating the amount of money she needs to repay a loan and the number of months she has been making payments.

A 2-column table with 5 rows. The first column is labeled months with entries 6, 12, 18, 24, 30. The second column is labeled amount to repay (dollar sign) with entries 2,700; 2,110; 1,110; 870; 220.
A graph shows months labeled 5 to 60 on the horizontal axis and amount to repay (dollar sign) on the vertical axis. A line shows a downward trend.

Answer 1

I'll fit a least-squares line to the five points (months x, amount y).

Data:
(6,2700), (12,2110), (18,1110), (24,870), (30,220).

Compute sums:
n = 5
sum x = 90
sum y = 7010
sum xy = 88,980
sum x^2 = 1,980

Slope:
m = (n·sum xy − sum x·sum y) / (n·sum x^2 − (sum x)^2)
= (5·88980 − 90·7010) / (5·1980 − 90^2)
= −186000 / 1800
= −103.333... ≈ −103.33

Intercept:
b = (sum y − m·sum x) / n
= (7010 − (−103.333..)*90) / 5
= 3262

Best-fit line:
Amount ≈ −103.33·(months) + 3262

Interpretation:
- The amount decreases by about $103.33 per month (slope).
- The model's estimate of the amount at month 0 (initial loan) is about $3,262 (intercept).
- Solve 0 = −103.33 x + 3262 → x ≈ 31.6 months, so the loan is predicted to be fully repaid after about 31.6 ≈ 32 months.