Question

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 188 cm. how close is the result to the actual opponent height of 175 cm? use a significance level of 0.05.
president 175 188 191 183 185 192 178 177 188
opponent 173 188 169 182 177 180 180 183 175
the regression equation is \\(\hat{y} = \square + (\square) x\\).
(round the y-intercept to the nearest integer as needed. round the slope to three decimal places as needed.)

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the following sums: \( n = 9 \) (number of pairs), \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \).

• \( \sum x = 175 + 188 + 191 + 183 + 185 + 192 + 178 + 177 + 188 = 1667 \)
• \( \sum y = 173 + 188 + 169 + 182 + 177 + 180 + 180 + 183 + 175 = 1607 \)
• \( \sum xy = (175×173)+(188×188)+(191×169)+(183×182)+(185×177)+(192×180)+(178×180)+(177×183)+(188×175) \)
• \( 175×173 = 30275 \)
• \( 188×188 = 35344 \)
• \( 191×169 = 32279 \)
• \( 183×182 = 33306 \)
• \( 185×177 = 32745 \)
• \( 192×180 = 34560 \)
• \( 178×180 = 32040 \)
• \( 177×183 = 32391 \)
• \( 188×175 = 32900 \)
• Sum: \( 30275 + 35344 + 32279 + 33306 + 32745 + 34560 + 32040 + 32391 + 32900 = 295840 \)
• \( \sum x^2 = 175^2 + 188^2 + 191^2 + 183^2 + 185^2 + 192^2 + 178^2 + 177^2 + 188^2 \)
• \( 175^2 = 30625 \)
• \( 188^2 = 35344 \)
• \( 191^2 = 36481 \)
• \( 183^2 = 33489 \)
• \( 185^2 = 34225 \)
• \( 192^2 = 36864 \)
• \( 178^2 = 31684 \)
• \( 177^2 = 31329 \)
• \( 188^2 = 35344 \)
• Sum: \( 30625 + 35344 + 36481 + 33489 + 34225 + 36864 + 31684 + 31329 + 35344 = 305385 \)

Step2: Calculate slope (\( b \)) and y-intercept (\( a \))

The formula for the slope \( b \) is:

$$ b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} $$

The formula for the y-intercept \( a \) is:

$$ a = \bar{y} - b\bar{x} $$

where \( \bar{x} = \frac{\sum x}{n} \) and \( \bar{y} = \frac{\sum y}{n} \).

• Calculate \( \bar{x} = \frac{1667}{9} \approx 185.2222 \)
• Calculate \( \bar{y} = \frac{1607}{9} \approx 178.5556 \)
• Calculate numerator for \( b \): \( 9×295840 - 1667×1607 \)
• \( 9×295840 = 2662560 \)
• \( 1667×1607 = 1667×(1600 + 7) = 1667×1600 + 1667×7 = 2667200 + 11669 = 2678869 \)
• Numerator: \( 2662560 - 2678869 = -16309 \)
• Calculate denominator for \( b \): \( 9×305385 - (1667)^2 \)
• \( 9×305385 = 2748465 \)
• \( (1667)^2 = 2778889 \)
• Denominator: \( 2748465 - 2778889 = -30424 \)
• Slope \( b = \frac{-16309}{-30424} \approx 0.536 \) (rounded to three decimal places)
• Calculate \( a = 178.5556 - 0.536×185.2222 \)
• \( 0.536×185.2222 \approx 99.28 \)
• \( a = 178.5556 - 99.28 \approx 79.2756 \approx 79 \) (rounded to nearest integer)

Answer:

The regression equation is \( \hat{y} = 79 + 0.536x \).