Question

1. the table below shows the number of doctors in bingham city from 1960-1986.

year	1960	1967	1970	1975	1982	1985	1986

a. predict the number of doctors in 1999. ________
b. predict the number of doctors in 2012. ________

Explanation:

Step1: Analyze the trend (linear regression)

First, we assign years as \( x \) (let 1960 be \( x = 0 \), 1967 be \( x = 7 \), 1970 be \( x = 10 \), 1975 be \( x = 15 \), 1982 be \( x = 22 \), 1985 be \( x = 25 \), 1986 be \( x = 26 \)) and number of doctors as \( y \).
We calculate the slope \( m \) and intercept \( b \) for linear regression. The formula for linear regression is \( y = mx + b \), where \( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \) and \( b=\frac{\sum y - m\sum x}{n} \).
First, list the values:
\( n = 7 \)
\( \sum x = 0 + 7 + 10 + 15 + 22 + 25 + 26 = 105 \)
\( \sum y = 2937 + 3511 + 3754 + 4173 + 4741 + 5019 + 5102 = 29237 \)
\( \sum xy = 0\times2937 + 7\times3511 + 10\times3754 + 15\times4173 + 22\times4741 + 25\times5019 + 26\times5102 \)
\( = 0 + 24577 + 37540 + 62595 + 104302 + 125475 + 132652 = 487141 \)
\( \sum x^{2} = 0^{2}+7^{2}+10^{2}+15^{2}+22^{2}+25^{2}+26^{2}=0 + 49 + 100 + 225 + 484 + 625 + 676 = 2159 \)

Calculate \( m \):
\( m=\frac{7\times487141 - 105\times29237}{7\times2159 - 105^{2}}=\frac{3409987 - 3069885}{15113 - 11025}=\frac{340102}{4088}\approx83.2 \)
Calculate \( b \):
\( b=\frac{29237 - 83.2\times105}{7}=\frac{29237 - 8736}{7}=\frac{20501}{7}\approx2928.7 \)
So the linear model is \( y = 83.2x + 2928.7 \)

Step2: Predict for 1999 (a)

For 1999, \( x = 1999 - 1960 = 39 \)
\( y = 83.2\times39 + 2928.7 = 3244.8 + 2928.7 = 6173.5\approx6174 \)

Step3: Predict for 2012 (b)

For 2012, \( x = 2012 - 1960 = 52 \)
\( y = 83.2\times52 + 2928.7 = 4326.4 + 2928.7 = 7255.1\approx7255 \)

Answer:

a. \(\boxed{6174}\) (approximate, depending on regression precision)
b. \(\boxed{7255}\) (approximate, depending on regression precision)