The Ipod Touch has been out for several years now and a lot of data has been collected.

There is a functional relationship between the Price of an IPod Touch and Weekly Demand.
Below is a table of data that have been collected

Price
P
($) Weekly Demand
S
(1,000s)
150 210
170 203
190 200
210 191
230 183
250 171

A.. Find the linear model that best fits this data using regression and enter the model below
(for entry round the slope value to nearest hundredth and constant parameter to nearest 1)
T
(
p
)
=

Now answer these two questions:

B.. What does the model predict will be the weekly demand if the price of an ipod touch is $220 ?
185000
Correct (nearest 100)

C.. According to the model at what should the price be set in order to have a weekly demand of 245,000 ipod Touches? $
(nearest $1)

Note that the first answer for parts B and C has been calculated using the rounded model parameters, whereas the second answer in both parts has been calculated using the unrounded model parameters.

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Step-by-step explanation:

The given table of data shows the functional relationship between the price (p) of an IPod Touch and the weekly demand (s):

[tex]\begin{array}{|c|c|}\cline{1-2} \textsf{Price, $p$, (\$)} & \textsf{Weekly Demand, $s$, (1,000s)} \\\cline{1-2} 150 & 210 \\\cline{1-2} 170 & 203 \\\cline{1-2} 190 & 200 \\\cline{1-2} 210 & 191 \\\cline{1-2} 230 & 183 \\\cline{1-2} 250 & 171 \\\cline{1-2} \end{array}[/tex]

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Part A

To find the linear model that best fits this data using regression, enter the data into a linear regression calculator, where the price (p) is the independent variable, and the weekly demand (s) is the dependent variable.

After entering the data from the given table into a statistical calculator we get:

[tex]a = -0.37714285...=-0.38[/tex]

[tex]b = 268.428571...=268[/tex]

where a is the slope (rounded to the nearest hundredth) and b is the y-intercept (rounded to the nearest 1) of a linear equation.

Therefore, the linear regression model is:

[tex]s=T(p)=\boxed{-0.38p+268}[/tex]

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Part B

To find the weekly demand if the price of an ipod touch is $220, substitute p = 220 into the regression model and solve for s.

Using the ROUNDED parameters:

[tex]s=T(220)=-0.38(220)+268\\\\s=T(220)=-83.6+268\\\\s=T(220)=184.4[/tex]

Using the UNROUNDED parameters:

[tex]s=T(220) = -0.37714285...(220) + 268.428571...\\\\s=T(220) =185.457142857142...\\\\s=T(220)=185.457[/tex]

Therefore, if the price of an ipod touch is $220, the weekly demand will be:

ROUNDED parameters = 184,400
UNROUNDED parameters = 185,500

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Part C

To determine the price at which the ipod should be set to have a weekly demand of 245,000, substitute s = 245 into the regression model and solve for p:

Using the ROUNDED parameters:

[tex]-0.38p+268=245\\\\\\-0.38p=-23\\\\\\p=\dfrac{-23}{-0.38}\\\\\\p=60.526315789...\\\\\\p=61[/tex]

Using the UNROUNDED parameters:

[tex]-0.37714285...(p) + 268.428571...=245\\\\p=62.12121212...\\\\p=62[/tex]

Therefore, to achieve a weekly demand of 245,000, the ipod should be set at a price of:

ROUNDED parameters = $61
UNROUNDED parameters = $62

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Additional Information

You have not specified if you are to use the ROUNDED or the UNROUNDED model parameters to answer parts B and C, so I have provided both.