Question

1. the data in the table to the left shows sales for a certain department store (in billions of dollars).

a) find the line of best fit:

b) estimate the stores sales in 2018.

year | 1994 | 1995 | 1996 | 1997 | 1998 | 1999
sales | 216 | 235 | 252 | 267 | 282 | 300

Explanation:

Step1: Assign Variables

Let \( x \) be the year (with \( x = 0 \) for 1994, \( x = 1 \) for 1995, ..., \( x = 5 \) for 1999) and \( y \) be sales (in billions). The data points are: \((0, 216)\), \((1, 235)\), \((2, 252)\), \((3, 267)\), \((4, 282)\), \((5, 300)\).

Step2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \) (\(\bar{x}\)): \(\frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5\)
Mean of \( y \) (\(\bar{y}\)): \(\frac{216 + 235 + 252 + 267 + 282 + 300}{6} = \frac{1552}{6} \approx 258.67\)

Step3: Calculate Slope (\( m \)) of Line of Best Fit

Slope formula: \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
Calculate numerator:
\((0 - 2.5)(216 - 258.67) + (1 - 2.5)(235 - 258.67) + (2 - 2.5)(252 - 258.67) + (3 - 2.5)(267 - 258.67) + (4 - 2.5)(282 - 258.67) + (5 - 2.5)(300 - 258.67)\)
\(= (-2.5)(-42.67) + (-1.5)(-23.67) + (-0.5)(-6.67) + (0.5)(8.33) + (1.5)(23.33) + (2.5)(41.33)\)
\(= 106.675 + 35.505 + 3.335 + 4.165 + 34.995 + 103.325 = 287.995 \approx 288\)

Calculate denominator:
\((0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2\)
\(= 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5\)

Thus, \( m = \frac{288}{17.5} \approx 16.46 \approx 16.5 \)

Step4: Calculate Y-Intercept (\( b \))

Using \( \bar{y} = m\bar{x} + b \):
\( 258.67 = 16.46(2.5) + b \)
\( 258.67 = 41.15 + b \)
\( b = 258.67 - 41.15 \approx 217.52 \approx 217.5 \)

Step5: Line of Best Fit

The line of best fit is \( y = 16.46x + 217.52 \) (or \( y \approx 16.5x + 217.5 \)).

Step6: Estimate Sales in 2018

For 2018, \( x = 2018 - 1994 = 24 \).
Substitute \( x = 24 \) into the line: \( y = 16.46(24) + 217.52 \)
\( y = 395.04 + 217.52 = 612.56 \) (or with \( y = 16.5x + 217.5 \): \( y = 16.5(24) + 217.5 = 396 + 217.5 = 613.5 \))

Part (a) Answer:

The line of best fit is approximately \( \boldsymbol{y = 16.5x + 217.5} \) (or \( y \approx 16.46x + 217.52 \)).

Part (b) Answer:

The estimated sales in 2018 are approximately \(\boldsymbol{613}\) (or 612.56) billion dollars.

Answer:

Step1: Assign Variables

Let \( x \) be the year (with \( x = 0 \) for 1994, \( x = 1 \) for 1995, ..., \( x = 5 \) for 1999) and \( y \) be sales (in billions). The data points are: \((0, 216)\), \((1, 235)\), \((2, 252)\), \((3, 267)\), \((4, 282)\), \((5, 300)\).

Step2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \) (\(\bar{x}\)): \(\frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5\)
Mean of \( y \) (\(\bar{y}\)): \(\frac{216 + 235 + 252 + 267 + 282 + 300}{6} = \frac{1552}{6} \approx 258.67\)

Step3: Calculate Slope (\( m \)) of Line of Best Fit

Slope formula: \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
Calculate numerator:
\((0 - 2.5)(216 - 258.67) + (1 - 2.5)(235 - 258.67) + (2 - 2.5)(252 - 258.67) + (3 - 2.5)(267 - 258.67) + (4 - 2.5)(282 - 258.67) + (5 - 2.5)(300 - 258.67)\)
\(= (-2.5)(-42.67) + (-1.5)(-23.67) + (-0.5)(-6.67) + (0.5)(8.33) + (1.5)(23.33) + (2.5)(41.33)\)
\(= 106.675 + 35.505 + 3.335 + 4.165 + 34.995 + 103.325 = 287.995 \approx 288\)

Calculate denominator:
\((0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2\)
\(= 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5\)

Thus, \( m = \frac{288}{17.5} \approx 16.46 \approx 16.5 \)

Step4: Calculate Y-Intercept (\( b \))

Using \( \bar{y} = m\bar{x} + b \):
\( 258.67 = 16.46(2.5) + b \)
\( 258.67 = 41.15 + b \)
\( b = 258.67 - 41.15 \approx 217.52 \approx 217.5 \)

Step5: Line of Best Fit

The line of best fit is \( y = 16.46x + 217.52 \) (or \( y \approx 16.5x + 217.5 \)).

Step6: Estimate Sales in 2018

For 2018, \( x = 2018 - 1994 = 24 \).
Substitute \( x = 24 \) into the line: \( y = 16.46(24) + 217.52 \)
\( y = 395.04 + 217.52 = 612.56 \) (or with \( y = 16.5x + 217.5 \): \( y = 16.5(24) + 217.5 = 396 + 217.5 = 613.5 \))

Part (a) Answer:

The line of best fit is approximately \( \boldsymbol{y = 16.5x + 217.5} \) (or \( y \approx 16.46x + 217.52 \)).

Part (b) Answer:

The estimated sales in 2018 are approximately \(\boldsymbol{613}\) (or 612.56) billion dollars.