Solved on Dec 05, 2023

Find the best predicted value of the response variable for x=3.5x=3.5 given r=0.742r=0.742 and the regression equation y^=55.8+2.79x\hat{y}=55.8+2.79x. Round to two decimal places. Use a significance level of 0.05.

STEP 1

Assumptions
1. The given regression equation is y^=55.8+2.79x\hat{y}=55.8+2.79x.
2. The correlation coefficient is r=0.742r=0.742.
3. The mean of the response variable yy is yˉ=71.125\bar{y}=71.125.
4. We are asked to predict the value of yy for x=3.5x=3.5.
5. The significance level is 0.050.05.
6. The critical values of the Pearson correlation coefficient rr are available if needed for determining the significance of the regression.

STEP 2

To find the best predicted value of the response variable yy for a given value of xx, we use the regression equation.
y^=a+bx\hat{y} = a + bx
where aa is the y-intercept and bb is the slope of the regression line.

STEP 3

Plug in the given value of x=3.5x=3.5 into the regression equation to calculate the predicted value of yy.
y^=55.8+2.79(3.5)\hat{y} = 55.8 + 2.79(3.5)

STEP 4

Calculate the value inside the parentheses first.
2.79×3.5=9.7652.79 \times 3.5 = 9.765

STEP 5

Now add this value to the y-intercept to find the predicted value of yy.
y^=55.8+9.765\hat{y} = 55.8 + 9.765

STEP 6

Calculate the sum to find the predicted value of yy.
y^=55.8+9.765=65.565\hat{y} = 55.8 + 9.765 = 65.565

STEP 7

Round the predicted value of yy to two decimal places.
y^65.57\hat{y} \approx 65.57
The best predicted value of yy for x=3.5x=3.5 is approximately 65.5765.57.

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