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The number of newly reported crime cases in a county in New York State is shown in the accompanying table, where [tex]$x$[/tex] represents the number of years since 2014, and [tex]$y$[/tex] represents the number of new cases. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the projected number of new cases for 2026, rounded to the nearest whole number.

\begin{tabular}{|c|c|}
\hline
Years since 2014 [tex][tex]$(x)$[/tex][/tex] & New Cases [tex]$(y)$[/tex] \\
\hline
0 & 1047 \\
\hline
1 & 1020 \\
\hline
2 & 989 \\
\hline
3 & 943 \\
\hline
4 & 964 \\
\hline
5 & 880 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's go through the steps to solve the problem step-by-step.

### Step 1: Organize the Data
We have the following data points:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1047 \\ \hline 1 & 1020 \\ \hline 2 & 989 \\ \hline 3 & 943 \\ \hline 4 & 964 \\ \hline 5 & 880 \\ \hline \end{array} \][/tex]

### Step 2: Calculate the Means
First, we calculate the mean of x and y:

[tex]\[ \bar{x} = \frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5 \][/tex]

[tex]\[ \bar{y} = \frac{1047 + 1020 + 989 + 943 + 964 + 880}{6} = \frac{5843}{6} \approx 973.8 \][/tex]

### Step 3: Calculate the Slope (m) and y-Intercept (b)

The formula for the slope [tex]\( m \)[/tex] of the least squares regression line is:

[tex]\[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \][/tex]

Let's calculate each part step-by-step.

#### Numerator Calculation:

[tex]\[ \sum{(x_i - \bar{x})(y_i - \bar{y})} \][/tex]

[tex]\[ = (0 - 2.5)(1047 - 973.8) + (1 - 2.5)(1020 - 973.8) + (2 - 2.5)(989 - 973.8) + (3 - 2.5)(943 - 973.8) + (4 - 2.5)(964 - 973.8) + (5 - 2.5)(880 - 973.8) \][/tex]

[tex]\[ = (-2.5 \times 73.2) + (-1.5 \times 46.2) + (-0.5 \times 15.2) + (0.5 \times (-30.8)) + (1.5 \times (-9.8)) + (2.5 \times (-93.8)) \][/tex]

[tex]\[ = -183 + -69.3 + -7.6 + -15.4 + -14.7 + -234.5 = -524.5 \][/tex]

#### Denominator Calculation:

[tex]\[ \sum{(x_i - \bar{x})^2} = (0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2 \][/tex]

[tex]\[ = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \][/tex]

#### Slope (m):

[tex]\[ m = \frac{-524.5}{17.5} \approx -30 \][/tex]

#### y-Intercept (b):

[tex]\[ b = \bar{y} - m\bar{x} \][/tex]

[tex]\[ b = 973.8 - (-30 \times 2.5) = 973.8 + 75 = 1048.8 \][/tex]

### Step 4: Write the Linear Regression Equation

The linear regression equation is:

[tex]\[ y = mx + b \][/tex]

Substituting the calculated [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:

[tex]\[ y = -30x + 1048.8 \][/tex]

Round the coefficients to the nearest tenth:

[tex]\[ y = -30x + 1048.8 \][/tex]

### Step 5: Project the Number of New Cases for 2026

To find the number of new cases for 2026, we need to calculate [tex]\( x \)[/tex] for the year 2026.

Since [tex]\( x \)[/tex] represents the number of years since 2014:

[tex]\[ x = 2026 - 2014 = 12 \][/tex]

Substitute [tex]\( x = 12 \)[/tex] into the linear regression equation:

[tex]\[ y = -30(12) + 1048.8 \][/tex]

[tex]\[ y = -360 + 1048.8 \][/tex]

[tex]\[ y = 688.8 \][/tex]

Rounding to the nearest whole number, the projected number of new cases for 2026 is:

[tex]\[ \boxed{689} \][/tex]

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