Questions: Listed below are the heights (cm) of winning presidential candidates and their main opponents from several recent presidential elections. Find the regression equation, letting president be the predictor (x) variable. Find the best predicted height of an opponent given that the president had a height of 177 cm. How close is the result to the actual opponent height of 183 cm? Use a significance level of 0.05. President: 183, 188, 183, 177, 192, 191, 178, 188, 175 Opponent: 185, 173, 182, 183, 180, 169, 180, 175, 173 The regression equation is ŷ = + ( x. x. (Round the y-intercept to the nearest integer as needed. Round the slope to three decimal places as needed.)

Solution Steps

To solve this problem, we need to perform a linear regression analysis where the height of the president is the predictor variable (x) and the height of the opponent is the response variable (y). The steps are as follows:

1. Import the necessary libraries.
2. Define the data for the heights of presidents and opponents.
3. Use a linear regression model to fit the data.
4. Extract the regression equation coefficients (slope and intercept).
5. Predict the opponent's height given a president's height of 177 cm.
6. Compare the predicted height to the actual height of 183 cm.

First, we need to calculate the means of the heights of the presidents and the opponents.

Given data:

• Presidents' heights: \(183, 188, 183, 177, 192, 191, 178, 188, 175\)
• Opponents' heights: \(185, 173, 182, 183, 180, 169, 180, 175, 173\)

Calculate the mean of the presidents' heights (\(\bar{x}\)): \[ \bar{x} = \frac{183 + 188 + 183 + 177 + 192 + 191 + 178 + 188 + 175}{9} = \frac{1655}{9} \approx 183.8889 \]

Calculate the mean of the opponents' heights (\(\bar{y}\)): \[ \bar{y} = \frac{185 + 173 + 182 + 183 + 180 + 169 + 180 + 175 + 173}{9} = \frac{1600}{9} \approx 177.7778 \]

Next, we calculate the slope (\(b\)) of the regression line using the formula: \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]

Calculate the numerator: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (183 - 183.8889)(185 - 177.7778) + (188 - 183.8889)(173 - 177.7778) + \ldots + (175 - 183.8889)(173 - 177.7778) \]

\[ = (-0.8889 \times 7.2222) + (4.1111 \times -4.7778) + (-0.8889 \times 4.2222) + (-6.8889 \times 5.2222) + (8.1111 \times 2.2222) + (7.1111 \times -8.7778) + (-5.8889 \times 2.2222) + (4.1111 \times -2.7778) + (-8.8889 \times -4.7778) \]

\[ = -6.4197 - 19.6395 - 3.7555 - 35.9605 + 18.0247 - 62.4444 - 13.0889 - 11.4167 + 42.4444 \]

\[ = -92.2561 \]

Calculate the denominator: \[ \sum (x_i - \bar{x})^2 = (183 - 183.8889)^2 + (188 - 183.8889)^2 + \ldots + (175 - 183.8889)^2 \]

\[ = (-0.8889)^2 + (4.1111)^2 + (-0.8889)^2 + (-6.8889)^2 + (8.1111)^2 + (7.1111)^2 + (-5.8889)^2 + (4.1111)^2 + (-8.8889)^2 \]

\[ = 0.7901 + 16.9012 + 0.7901 + 47.4412 + 65.7877 + 50.5667 + 34.6667 + 16.9012 + 79.0111 \]

\[ = 312.8560 \]

Now, calculate the slope: \[ b = \frac{-92.2561}{312.8560} \approx -0.295 \]

The y-intercept (\(a\)) is calculated using the formula: \[ a = \bar{y} - b\bar{x} \]

\[ a = 177.7778 - (-0.295 \times 183.8889) \]

\[ a = 177.7778 + 54.2222 \approx 232 \]

The regression equation is: \[ \hat{y} = a + bx \]

Substitute the values of \(a\) and \(b\): \[ \hat{y} = 232 - 0.295x \]

Given that the president's height (\(x\)) is 177 cm, we can predict the opponent's height (\(\hat{y}\)):

\[ \hat{y} = 232 - 0.295 \times 177 \]

\[ \hat{y} = 232 - 52.215 \approx 179.785 \]

The actual opponent height is 183 cm. The difference between the predicted and actual height is:

\[ |183 - 179.785| = 3.215 \]

Final Answer