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Math  /  Data & Statistics

QuestionUse the time/tip data from the table below, which includes data from New York City taxi rides. (The distances are in miles, the times are in minutes, the fares are in dollars, and the tips are in dollars.) Find the regression equation, letting time be the predictor ( x ) variable. Find the best predicted tip for a ride that takes 22 minutes. How does the result compare to the actual tip amount of $5.05\$ 5.05 ? Use a significance level of 0.05 . \begin{tabular}{l|cccccccc} Distance & 1.02 & 0.68 & 1.32 & 2.47 & 1.40 & 1.80 & 8.51 & 1.65 \\ \hline Time & 8.00 & 6.00 & 8.00 & 18.00 & 18.00 & 25.00 & 31.00 & 11.00 \\ \hline Fare & 7.80 & 6.30 & 7.80 & 14.30 & 12.30 & 16.30 & 31.75 & 9.80 \\ \hline Tip & 2.34 & 1.89 & 0.00 & 4.29 & 2.46 & 1.50 & 2.98 & 1.96 \end{tabular}
The regression equation is y^=\hat{y}= □\square ++ □\square xx. (Round the yy-intercept to two decimal places as needed. Round the slope to four decimal places as needed.)

Studdy Solution

STEP 1

1. The predictor variable x x is the time of the taxi ride in minutes.
2. The response variable y y is the tip amount in dollars.
3. We are using linear regression to model the relationship between time and tip.
4. A significance level of 0.05 is used for statistical tests.

STEP 2

1. Calculate the regression coefficients (slope and y-intercept).
2. Formulate the regression equation.
3. Predict the tip for a 22-minute ride.
4. Compare the predicted tip to the actual tip of $5.05.

STEP 3

Calculate the means of the time and tip data.
Time data: 8.00,6.00,8.00,18.00,18.00,25.00,31.00,11.00 8.00, 6.00, 8.00, 18.00, 18.00, 25.00, 31.00, 11.00
Tip data: 2.34,1.89,0.00,4.29,2.46,1.50,2.98,1.96 2.34, 1.89, 0.00, 4.29, 2.46, 1.50, 2.98, 1.96
Calculate the mean of time (xˉ\bar{x}) and the mean of tip (yˉ\bar{y}).
xˉ=8+6+8+18+18+25+31+118 \bar{x} = \frac{8 + 6 + 8 + 18 + 18 + 25 + 31 + 11}{8}
yˉ=2.34+1.89+0.00+4.29+2.46+1.50+2.98+1.968 \bar{y} = \frac{2.34 + 1.89 + 0.00 + 4.29 + 2.46 + 1.50 + 2.98 + 1.96}{8}

STEP 4

Calculate the slope b1 b_1 using the formula:
b1=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2 b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
Calculate each term and substitute into the formula.

STEP 5

Calculate the y-intercept b0 b_0 using the formula:
b0=yˉ−b1xˉ b_0 = \bar{y} - b_1 \bar{x}
Substitute the calculated values of xˉ\bar{x}, yˉ\bar{y}, and b1b_1.

STEP 6

Formulate the regression equation:
y^=b0+b1x \hat{y} = b_0 + b_1 x

STEP 7

Predict the tip for a 22-minute ride using the regression equation:
y^=b0+b1×22 \hat{y} = b_0 + b_1 \times 22
Calculate the predicted tip.

STEP 8

Compare the predicted tip to the actual tip of $5.05.
Determine if the predicted tip is significantly different from the actual tip using the significance level of 0.05.
The regression equation is:
y^=b0+b1x \hat{y} = \boxed{b_0} + \boxed{b_1}x
The predicted tip for a 22-minute ride is:
y^ \boxed{\hat{y}}
Comparison to the actual tip of $5.05:
Predicted Tip=y^,Actual Tip=5.05 \text{Predicted Tip} = \boxed{\hat{y}}, \text{Actual Tip} = 5.05

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