Café Michigan's manager, Gary Stark, suspects that demand for mocha latte coffees depends on the price being charged. Based on historical observations, Gary has gathered the following data, which show the numbers of these coffees sold over six different price values:

Price Number Sold

$2.70 760
$3.50 510
$2.00 980
$4.20 250
$3.10 320
$4.05 480

Using these data, how many mocha latte coffees would be forecast to be sold according to simple linear regression if the price per cup were $2.80?

Can you please show your work?

677 units

At $2.80, the forecast for sales is 677 units

Explanation:

No           Price (x)     Number Sold (y)          xy        x²         y²

                    ($)                                   ($)          ($)         ($)

1                  2.7          760                2052        7.29     577600

2                 3.5          510                         1785       12.25    260100

3                   2                  980                 19604     960400

4                 4.2          250                 1050      17.64     62500

5                 3.1          320                 992  9.61      102400

6                4.05         480                       1944    16.4025     230400

∑                19.55        3300                9783    67.1925    2,193,400

Y = a + bX

Where a = {(∑y)(∑x²) - (∑x)(∑xy)} / {n(∑x²) - (∑x)²}

b = {n(∑xy) - (∑x)(∑y)} / {n(∑x²) - (∑x)²}

a = {(3300*67.1925) - (19.55*9783)} / {(6*67.1925) - 19.55²}

= {221,735.25‬ - 191,257.65} / {403.155 - 382.2025‬}

= 30,477.6‬/20.9525‬

= 1,454.60

b = {(6*9783) - (19.55*3300)} / 20.9525

= {58,698‬ - 64,515‬} / 20.9525

= -5,817‬/20.9525

= -277.62

putting the equation together:

Y = 1454.60 + (-277.62)X

∴ when x = $2.80

Y = 1454.60 - 277.62(2.80)

= 1454.60 - 777.34

= 677.26‬ ≈ 677 units

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