Question

What is the equation for the line of best fit for the following data? Round theslope and y-intercept of the line to three decimal places.X2571216y30192092

Answer 1

Tutancamon mathematics

Answer 2

Rounding to three decimal places, the slope is approximately -1.892 and the y-intercept is approximately 31.931. Therefore, the equation for the line of best fit is closest to: c. y = -1.893x + 31.901

To find the equation for the line of best fit, we'll use the least squares method to calculate the slope (m) and y-intercept (b).

Given data:

X: 2, 5, 7, 12, 16

y: 30, 19, 20, 9, 2

Let's first calculate the means of X and y:

`\[ \bar{X} = (2 + 5 + 7 + 12 + 16)/(5) = (42)/(5) = 8.4 \]`

`\[ \bar{y} = (30 + 19 + 20 + 9 + 2)/(5) = (80)/(5) = 16 \]`

Now, let's calculate the slope (m) using the formula:

`\[ m = \frac{\sum_(i=1)^(n) (x_i - \bar{X})(y_i - \bar{y})}{\sum_(i=1)^(n) (x_i - \bar{X})^2} \]`

Calculating the numerator and denominator:

Numerator:

`\[ (2 - 8.4)(30 - 16) + (5 - 8.4)(19 - 16) + (7 - 8.4)(20 - 16) + (12 - 8.4)(9 - 16) + (16 - 8.4)(2 - 16) \]`

`\[ (-6.4)(14) + (-3.4)(3) + (-1.4)(4) + (3.6)(-7) + (7.6)(-14) \]`

`\[ -89.6 - 10.2 - 5.6 - 25.2 - 106.4 = -237 \]`

Denominator:

`\[ (2 - 8.4)^2 + (5 - 8.4)^2 + (7 - 8.4)^2 + (12 - 8.4)^2 + (16 - 8.4)^2 \]`

`\[ (-6.4)^2 + (-3.4)^2 + (-1.4)^2 + (3.6)^2 + (7.6)^2 \]`

`\[ 40.96 + 11.56 + 1.96 + 12.96 + 57.76 = 125.2 \]`

Now, let's calculate the slope m:

`\[ m = (-237)/(125.2) \approx -1.892 \]`

Next, let's calculate the y-intercept b using the formula:

`\[ b = \bar{y} - m * \bar{X} \]`

`\[ b = 16 - (-1.892) * 8.4 \]`

`\[ b = 16 + 15.931 \]`

`\[ b \approx 31.931 \]`

Rounding to three decimal places, the slope is approximately -1.892 and the y-intercept is approximately 31.931. Therefore, the equation for the line of best fit is closest to option: c. y = -1.893x + 31.901