Question

I am having trouble with the final step in this assignment, "How many shirts per month will be demanded if the price is $27?"I believe I have solved correctly the prerequisite problems of 1-5.I would enjoy going over those and see if my math is correct and my problem solving is using the correct approaches.

Answer 1

1. The relationship is not a function because one input (22) has 2 different possible outputs (45 and 42).

2, 3 and 4:

The following graph shows the scatter plot, the best fit line and its equation:

The equation of the line, where x represents the price and y represents the demand.

Rewriting the equation we have:

`D=-1.3355\cdot P+73.862`

That equation is found in Excel. However it can be computed through the following equations,

The formula has the shape:

`y=mx+b`

The formula for the slope m:

`m=\frac{n\sum^{}_{}xy-\sum^{}_{}x\cdot\sum^{}_{}y}{n\sum^{}_{}x^2-(\sum^{}_{}x)^2}`

Where n is the number of data, 7 for this case.

For the y-intercept we have the following formula:

`b=\frac{\sum^{}_{}y}{n}-m\frac{\sum^{}_{}x}{n}`

x y xyx^2

19 49 931361

21 46 966441

22 45 990484

22 42 924484

26 41 1066676

28 38 1064784

29 33 957841

167 294 68984071 <- SUM

The previous table shows the same original table (P, D) with 2 new columns to calculate the sums required for the formula of the slope and y-intercept. The last row is the total sum of all the elements above in the collumn.

From the table we have then:

`\begin{gathered} \sum ^{}_{}x=167 \\ \sum ^{}_{}y=294 \\ \sum ^{}_{}xy=6898 \\ \sum ^{}_{}x^2=4071 \\ n=7 \end{gathered}`

Now we will replace the values on the previous formulas for m and b:

`m=\frac{n\sum^{}_{}xy-\sum^{}_{}x\cdot\sum^{}_{}y}{n\sum^{}_{}x^2-(\sum^{}_{}x)^2}=(7\cdot6898-167\cdot294)/(7\cdot4071-(167)^2)`

Solving we obtain:

`m=-(203)/(152)\approx-1.3355`

Now, for the y-intercept b:

`\begin{gathered} b=\frac{\sum^{}_{}y}{n}-m\frac{\sum^{}_{}x}{n}=(294)/(7)-(-(203)/(152))\cdot(167)/(7) \\ \\ b=(11227)/(152)\approx73.862 \end{gathered}`

Now we can build the equation of the best fit of the model. We used x and y to make the formulas easier to work, however, x represents the price (P) and y is the demand (D). Reorganizing then the notation:

`D=-(203)/(152)\cdot P+(11227)/(152)`

Or well:

`D=-1.3355\cdot P+73.862`

To make formal the answer for point 5:

`\text{Slope}=-(203)/(152)\approx-1.3355`
`\text{y-intercept}=(11227)/(152)\approx73.862`

The slope represents the rate of change. It means that for every unit the price increases, the demand will decrease 1.3355 (The negative means decrement).

The y-intercept will be the demand when the price is exactly 0. Then, according to that model, the demand of the shirts if they were free will be 73.862 approximately.

6. The equation as function D of P was already found:

`\begin{gathered} D=-(203)/(152)\cdot P+(11227)/(152) \\ \\ D=-1.3355\cdot P+73.862 \end{gathered}`

As a linear function, its domain will be all real numbers:

`\text{Domain}=(-\infty,\infty)`

7. To make the prediction of the demand if the price is $27, we just need to evaluate the equation we found in P = $27, and solve:

`\begin{gathered} D=-(203)/(152)\cdot(27)+(11227)/(152) \\ \\ D\approx37.8 \end{gathered}`

Then, the demand for shirts, if the price were $27, will be approximately 37.8, according to the model.