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 2009, 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 2014, rounded to the nearest whole number.
years since 2009 (x) | new cases (y)
0 | 1130
1 | 1172
2 | 1180
3 | 1279
4 | 1275
copy values for calculator
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answer
regression equation:
final answer:

Explanation:

Step 1: Calculate necessary sums

First, we need to calculate the sums for \(x\), \(y\), \(xy\), and \(x^2\).
Given the data points: \((0, 1130)\), \((1, 1172)\), \((2, 1180)\), \((3, 1279)\), \((4, 1275)\)

- \(n = 5\) (number of data points)
- \(\sum x = 0 + 1 + 2 + 3 + 4 = 10\)
- \(\sum y = 1130 + 1172 + 1180 + 1279 + 1275 = 5036\)
- \(\sum xy = (0\times1130)+(1\times1172)+(2\times1180)+(3\times1279)+(4\times1275)= 0 + 1172 + 2360 + 3837 + 5100 = 12469\)
- \(\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30\)

Step 2: Calculate slope (\(m\)) and y-intercept (\(b\))

The formula for the slope \(m\) of the linear regression line \(y = mx + b\) is:
\[m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2}\]
Substitute the values:
\[m=\frac{5\times12469 - 10\times5036}{5\times30 - 10^2}=\frac{62345 - 50360}{150 - 100}=\frac{11985}{50}=239.7\]
The formula for the y-intercept \(b\) is:
\[b=\frac{\sum y - m\sum x}{n}\]
Substitute the values:
\[b=\frac{5036 - 239.7\times10}{5}=\frac{5036 - 2397}{5}=\frac{2639}{5}=527.8\]
So the linear regression equation is \(y = 239.7x + 527.8\) (rounded to the nearest hundredth).

Step 3: Predict for 2011

For 2011, the number of years since 2009 (\(x\)) is \(2011 - 2009 = 2\)? Wait, no: 2009 is \(x = 0\), so 2010 is \(x = 1\), 2011 is \(x = 2\)? Wait, wait, 2009: \(x = 0\), 2010: \(x = 1\), 2011: \(x = 2\)? Wait, no, 2009 + 2 years is 2011, so \(x = 2\)? Wait, no, 2009 is \(x = 0\), so 2011 is \(x = 2011 - 2009 = 2\). Wait, but let's check the data: \(x = 0\) is 2009, \(x = 1\) is 2010, \(x = 2\) is 2011? Wait, the table has \(x = 0\) (2009), \(x = 1\) (2010), \(x = 2\) (2011), \(x = 3\) (2012), \(x = 4\) (2013). Wait, the problem says "for 2011", so \(x = 2\) (since 2009 is \(x = 0\), 2010 is \(x = 1\), 2011 is \(x = 2\)).

Wait, no, wait: 2009 is \(x = 0\), so the year is \(2009 + x\). So for 2011, \(x = 2011 - 2009 = 2\).

Now, substitute \(x = 2\) into the regression equation \(y = 239.7x + 527.8\)

\[y = 239.7\times2 + 527.8 = 479.4 + 527.8 = 1007.2\] Wait, that can't be right, because the original data for \(x = 2\) (2011?) Wait, no, the table has \(x = 2\) with \(y = 1180\). Wait, maybe I made a mistake in the year calculation. Wait, the problem says "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 2009, 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 2011, rounded to the nearest whole number."

Wait, 2009: \(x = 0\), 2010: \(x = 1\), 2011: \(x = 2\). But in the table, \(x = 2\) has \(y = 1180\). But we need to use the regression equation to predict for 2011. Wait, maybe my calculation of the regression equation is wrong. Let's recalculate the sums.

Wait, \(\sum y\): 1130 + 1172 = 2302; 2302 + 1180 = 3482; 3482 + 1279 = 4761; 4761 + 1275 = 6036? Wait, I think I added wrong earlier. 1130 + 1172 = 2302; 2302 + 1180 = 3482; 3482 + 1279 = 4761; 4761 + 1275 = 6036. Oh! I made a mistake in \(\sum y\) earlier. That's the error.

So let's recalculate:

- \(\sum x = 0 + 1 + 2 + 3 + 4 = 10\)
- \(\sum y = 1130 + 1172 + 1180 + 1279 + 1275 = 6036\) (correct sum)
- \(\sum xy = 0\times1130 + 1\times1172 + 2\times1180 + 3\times1279 + 4\times1275 = 0 + 1172 + 2360 + 3837 + 5100 = 12469\) (this was correct)
- \(\sum x^2 = 0 + 1 + 4 + 9 + 16 = 30\) (correct)

Now, recalculate \(m\):

\[m=\frac{5\times12469 - 10\times6036}{5\times30 - 10^2}=\frac{62345 - 60360}{150 - 100}=\frac{1985}{50}=39.7\]

Now, recalculate \(b\):

\[b=\frac{6036 - 39.7\times10}{5}=\frac{6036 - 397}{5}=\frac{5639}{5}=1127.8\]

So the linear regression equation is \(y = 39.7x + 1127.8\) (rounded to the nearest hundredth).

Now, for 2011, \(x = 2011 - 2009 = 2\). Substitute \(x = 2\) into the equation:

\[y = 39.7\times2 + 1127.8 = 79.4 + 1127.8 = 1207.2\]

Rounded to the nearest whole number, that's 1207.

Wait, let's verify the regression equation with the data. Let's check \(x = 0\): \(y = 1127.8\), which is close to 1130. \(x = 1\): \(39.7 + 1127.8 = 1167.5\), close to 1172. \(x = 2\): \(79.4 + 1127.8 = 1207.2\), while the actual \(y\) for \(x = 2\) is 1180. \(x = 3\): \(39.7\times3 + 1127.8 = 119.1 + 1127.8 = 1246.9\), close to 1279? Wait, no, 1246.9 vs 1279. \(x = 4\): \(39.7\times4 + 1127.8 = 158.8 + 1127.8 = 1286.6\), close to 1275. Hmm, maybe my calculation of \(\sum y\) was still wrong? Wait, 1130 + 1172 = 2302; 2302 + 1180 = 3482; 3482 + 1279 = 4761; 4761 + 1275 = 6036. That's correct. Wait, maybe the problem is that 2011 is \(x = 2\), but let's check the years again. If \(x = 0\) is 2009, then \(x = 1\) is 2010, \(x = 2\) is 2011, \(x = 3\) is 2012, \(x = 4\) is 2013. So the data points are for 2009, 2010, 2011, 2012, 2013? Wait, the table has \(x = 2\) with \(y = 1180\), which would be 2011. But the problem says "find the projected number of new cases for 2011", so maybe \(x = 2\) is 2011, but we are using the regression equation to predict, not the actual data. Wait, maybe my initial sum of \(y\) was wrong. Let's add again: 1130 (2009) + 1172 (2010) + 1180 (2011) + 1279 (2012) + 1275 (2013) = 1130 + 1172 = 2302; 2302 + 1180 = 3482; 3482 + 1279 = 4761; 4761 + 1275 = 6036. Correct.

Wait, maybe the regression equation is calculated using a calculator. Let's use the formula for linear regression:

The slope \(m\) is \(\frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)

\(n = 5\), \(\sum x = 10\), \(\sum y = 6036\), \(\sum xy = 12469\), \(\sum x^2 = 30\)

So numerator for \(m\): \(5*12469 - 10*6036 = 62345 - 60360 = 1985\)

Denominator: \(5*30 - 10^2 = 150 - 100 = 50\)

\(m = 1985 / 50 = 39.7\)

\(b = (\sum y - m\sum x)/n = (6036 - 39.7*10)/5 = (6036 - 397)/5 = 5639/5 = 1127.8\)

So the equation is \(y = 39.7x + 1127.8\)

Now, for 2011, \(x = 2\) (since 2009 is \(x=0\), 2011 is \(x=2\))

\(y = 39.7*2 + 1127.8 = 79.4 + 1127.8 = 1207.2\), which rounds to 1207.

Answer:

The linear regression equation is \(y = 39.70x + 1127.80\) (rounded to the nearest hundredth). The projected number of new cases for 2011 is \(\boxed{1207}\).