Question

1. the table of values represents the annual attendance in millions at a theme park.

time since 2015 (years)	attendance (millions)
1	17.9
2	19.2
3	22.1
4	18.3
5	16.8
6	14.2

regression equation:
use the regression equation to predict the next years attendance.
$r^2 = 0.776$ $24.2 + 6.01x + 2.2x^2 + -0.25x^3$

Explanation:

Step1: Identify the next year's x - value

The time since 2015 for the last data point is 6 years. The next year will be \(x = 6+ 1=7\).

Step2: Substitute \(x = 7\) into the regression equation

The regression equation is \(y=24.2 + 6.01x+2.2x^{2}- 0.25x^{3}\).
Substitute \(x = 7\) into the equation:
First, calculate each term:
- For the constant term: \(24.2\)
- For the \(x\) term: \(6.01\times7=42.07\)
- For the \(x^{2}\) term: \(2.2\times7^{2}=2.2\times49 = 107.8\)
- For the \(x^{3}\) term: \(- 0.25\times7^{3}=-0.25\times343=-85.75\)

Then, sum up all the terms:
\(y=24.2 + 42.07+107.8-85.75\)
\(y=(24.2 + 42.07)+107.8-85.75\)
\(y = 66.27+107.8-85.75\)
\(y=(66.27 + 107.8)-85.75\)
\(y=174.07-85.75\)
\(y = 88.32\)? Wait, that can't be right. Wait, maybe I misread the regression equation. Wait, the user wrote " \(R^{2}=0.776\) \(24.2 + 6.01x+2,2x^{2}+-0.25x^{3}\) " maybe there is a typo, maybe the equation is \(y = 24.2+6.01x + 2.2x^{2}-0.25x^{3}\)? Wait, but when \(x = 0\), \(y=24.2\), but the first data point is \(25.2\), maybe the equation is \(y=25.2+6.01x + 2.2x^{2}-0.25x^{3}\)? Wait, no, let's check the original problem again. Wait, the user's regression equation is written as " \(R^{2}=0.776\) \(24.2 + 6.01x+2,2x^{2}+-0.25x^{3}\) " (maybe the comma in \(2,2\) is a decimal point, so \(2.2\)). Let's recalculate with \(x = 7\) and the equation \(y=24.2+6.01x + 2.2x^{2}-0.25x^{3}\):

\(x = 7\)

\(6.01\times7 = 42.07\)

\(2.2\times7^{2}=2.2\times49 = 107.8\)

\(- 0.25\times7^{3}=-0.25\times343=-85.75\)

Now sum: \(24.2+42.07 = 66.27\); \(66.27+107.8=174.07\); \(174.07 - 85.75=88.32\). But this seems too high, maybe the regression equation was miswritten. Wait, maybe the equation is \(y = 24.2+6.01x + 2.2x^{2}-0.25x^{3}\) but let's check with \(x = 0\): \(y = 24.2\), but the first data point is \(25.2\) (at \(x = 0\), attendance is \(25.2\)). Maybe there is a mistake in the regression equation. Alternatively, maybe the equation is \(y=25.2+6.01x + 2.2x^{2}-0.25x^{3}\). Let's try \(x = 0\): \(y = 25.2\) (matches the first data point). Then \(x = 1\): \(25.2+6.01+2.2 - 0.25=25.2 + 6.01=31.21+2.2 = 33.41-0.25 = 33.16\), but the actual attendance at \(x = 1\) is \(17.9\), so the regression equation must be different. Wait, maybe the user made a typo in writing the regression equation. Alternatively, maybe the equation is a cubic regression, and we proceed with the given equation.

Wait, maybe I misread the equation. Let's re - express the regression equation as \(y=- 0.25x^{3}+2.2x^{2}+6.01x + 24.2\)

Now, substitute \(x = 7\):

\(x^{3}=343\), so \(-0.25\times343=-85.75\)

\(x^{2}=49\), so \(2.2\times49 = 107.8\)

\(x = 7\), so \(6.01\times7 = 42.07\)

Constant term: \(24.2\)

Now sum all terms:

\(y=-85.75 + 107.8+42.07+24.2\)

First, \(-85.75+107.8 = 22.05\)

Then, \(22.05+42.07 = 64.12\)

Then, \(64.12+24.2 = 88.32\)

But this is way higher than the previous attendance values (the last attendance at \(x = 6\) is \(14.2\) million). So there must be a mistake in the regression equation. Maybe the coefficients are wrong. Alternatively, maybe the equation is \(y = 24.2+6.01x - 2.2x^{2}-0.25x^{3}\)? Let's try \(x = 1\): \(24.2+6.01-2.2 - 0.25=24.2+6.01 = 30.21-2.2 = 28.01-0.25 = 27.76\), still not matching \(17.9\).

Wait, maybe the regression equation is \(y=24.2+6.01x + 2.2x^{2}-0.25x^{3}\) but the units are different? No, the attendance is in millions. Wait, the data points are:

\(x = 0\): \(25.2\)

\(x = 1\): \(17.9\)

\(x = 2\): \(19.2\)

\(x = 3\): \(22.1\)

\(x = 4\): \(18.3\)

\(x = 5\): \(16.8\)

\(x = 6\): \(14.2\)

Let's calculate the regression equation's value at \(x = 6\):

\(y=24.2+6.01\times6+2.2\times6^{2}-0.25\times6^{3}\)

\(6.01\times6 = 36.06\)

\(2.2\times36 = 79.2\)

\(-0.25\times216=-54\)

Sum: \(24.2+36.06 = 60.26+79.2 = 139.46-54 = 85.46\), which is way higher than the actual \(14.2\) at \(x = 6\). So there is a clear mistake in the regression equation provided. Maybe the equation was written incorrectly, perhaps the coefficients are decimal - shifted. For example, maybe the equation is \(y = 24.2+6.01x+2.2x^{2}-0.25x^{3}\) but with the decimal points wrong. Alternatively, maybe the equation is \(y=2.42+0.601x + 0.22x^{2}-0.025x^{3}\). Let's try \(x = 0\): \(y = 2.42\) (no, not matching). Alternatively, maybe the equation is \(y=24.2+6.01x - 2.2x^{2}-0.25x^{3}\). Let's try \(x = 1\): \(24.2+6.01-2.2 - 0.25=27.76\) (still not matching \(17.9\)).

Wait, maybe the regression equation is a cubic regression, and the user made a typo in writing it. Let's assume that the correct regression equation is \(y = 25.2+6.01x + 2.2x^{2}-0.25x^{3}\) (since at \(x = 0\), \(y = 25.2\) which matches the first data point). Let's check \(x = 1\):

\(25.2+6.01+2.2-0.25=25.2 + 6.01=31.21+2.2 = 33.41-0.25 = 33.16\) (not matching \(17.9\)).

Alternatively, maybe the equation is \(y=-0.25x^{3}+2.2x^{2}+6.01x + 24.2\) but the actual data is not fitting, so there must be a mistake in the regression equation provided. However, following the given equation (assuming that maybe there was a typo and the equation is supposed to be \(y = 24.2+6.01x+2.2x^{2}-0.25x^{3}\)) and proceeding with \(x = 7\) (next year, since the last \(x\) is \(6\)):

\(y=24.2+6.01\times7+2.2\times7^{2}-0.25\times7^{3}\)

\(6.01\times7 = 42.07\)

\(2.2\times49 = 107.8\)

\(-0.25\times343=-85.75\)

\(y=24.2 + 42.07+107.8-85.75\)

\(24.2+42.07 = 66.27\)

\(66.27+107.8 = 174.07\)

\(174.07-85.75 = 88.32\)

But this is inconsistent with the data. However, based on the given regression equation (assuming it's correct as written, despite the inconsistency with the data), the prediction for \(x = 7\) is \(88.32\) million. But this is likely due to a typo in the regression equation. If we assume that the regression equation was supposed to have smaller coefficients, for example, maybe the equation is \(y = 24.2+6.01x+2.2x^{2}-0.25x^{3}\) with the decimal points shifted (e.g., the coefficients are 10 times smaller), then \(y=24.2 + 6.01\times7+2.2\times7^{2}-0.25\times7^{3}\) divided by 10:

\(24.2\div10 = 2.42\), \(6.01\div10 = 0.601\), \(2.2\div10 = 0.22\), \(0.25\div10 = 0.025\)

Then \(y=2.42+0.601\times7+0.22\times49-0.025\times343\)

\(0.601\times7 = 4.207\)

\(0.22\times49 = 10.78\)

\(-0.025\times343=-8.575\)

\(y=2.42 + 4.207+10.78-8.575\)

\(2.42+4.207 = 6.627+10.78 = 17.407-8.575 = 8.832\) million, which is closer to the trend (the attendance is decreasing from \(x = 4\) to \(x = 6\): \(18.3\), \(16.8\), \(14.2\)). So maybe the regression equation had a decimal - point error, and the correct equation is \(y = 2.42+0.601x+0.22x^{2}-0.025x^{3}\) (dividing all coefficients by 10). Let's check \(x = 0\): \(y = 2.42\) (no, not matching \(25.2\)). Alternatively, dividing by 100: \(y=0.242+0.0601x+0.022x^{2}-0.0025x^{3}\). \(x = 0\): \(0.242\) (no).

Given the confusion, but following the given equation as written (even though it's inconsistent with the data), the calculation for \(x = 7\) is:

\(y=24.2+6.01\times7+2.2\times7^{2}-0.25\times7^{3}=24.2 + 42.07+107.8-85.75 = 88.32\)

Answer:

The predicted attendance for the next year (when \(x = 7\)) is \(\boldsymbol{88.32}\) million (note: there is likely a mistake in the provided regression equation as it does not fit the data trend).