Question

The number of newly reported crime cases in a county in New York State is shown in the accompanying table, where x represents the number of years since 1998, and y represents number of new cases. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the projected number of new cases for 2006, rounded to the nearest whole number.

Answer 1

The linear regression equation for the number of newly reported crime cases is y = 11.65x + 1127.25, where x is the number of years since 1998. The projected cases for 2006 is approximately 1222.

To find the linear regression equation, we can use the least squares method. The general form of the equation is y = mx + b, where m is the slope and b is the y-intercept.

Using the given data points (0, 1129), (1, 1126), (2, 1149), (3, 1194), and (4, 1197), the calculations for the slope (m) and y-intercept (b) are as follows:

`\[ m = (n(\sum xy) - (\sum x)(\sum y))/(n(\sum x^2) - (\sum x)^2) \]\[ b = ((\sum y) - m(\sum x))/(n) \]`

After calculating, the rounded coefficients are
`\(m \approx 11.65\)` and
`\(b \approx 1127.25\)`. The linear regression equation is then y = 11.65x + 1127.25.

To find the projected number of new cases for 2006 (8 years since 1998), substitute x = 8 into the equation:

`\[ y = 11.65 * 8 + 1127.25 \]`

Calculating this gives
`\(y \approx 1222\)`. Therefore, the projected number of new cases for 2006 is approximately 1222.