Question

situation: the ipod touch released in 2019 and a lot of data has been collected. relevant relationship: there is a functional relationship between price of an ipod touch, p and weekly demand, s. below is a table of data that have been collected price, p, ($) | weekly demand, s, (1,000s) 150 | 212 170 | 208 190 | 195 210 | 191 230 | 180 250 | 170 find the linear model that best fits this data using regression and enter the model below (round to the nearest hundredths) s = t(p) = question help: written example message instructor

Explanation:

Step1: Define variables and formulas

Let \(p\) = price (independent variable), \(s\) = weekly demand (dependent variable). The linear model is \(s = mp + b\), where:
$$m = \frac{n\sum(p_i s_i) - \sum p_i \sum s_i}{n\sum p_i^2 - (\sum p_i)^2}$$
$$b = \frac{\sum s_i - m\sum p_i}{n}$$
\(n=6\) (number of data points)

Step2: Calculate required sums

First, compute each sum from the table:

• \(\sum p_i = 150+170+190+210+230+250 = 1200\)
• \(\sum s_i = 212+208+195+191+180+170 = 1156\)
• \(\sum(p_i s_i) = (150×212)+(170×208)+(190×195)+(210×191)+(230×180)+(250×170) = 31800+35360+37050+40110+41400+42500 = 228220\)
• \(\sum p_i^2 = 150^2+170^2+190^2+210^2+230^2+250^2 = 22500+28900+36100+44100+52900+62500 = 247000\)

Step3: Calculate slope \(m\)

Substitute values into the slope formula:
$$m = \frac{6×228220 - 1200×1156}{6×247000 - (1200)^2}$$
$$= \frac{1369320 - 1387200}{1482000 - 1440000}$$
$$= \frac{-17880}{42000} ≈ -0.4257 ≈ -0.43$$

Step4: Calculate intercept \(b\)

Substitute \(m\), \(\sum p_i\), \(\sum s_i\) into the intercept formula:
$$b = \frac{1156 - (-0.4257)×1200}{6}$$
$$= \frac{1156 + 510.84}{6}$$
$$= \frac{1666.84}{6} ≈ 277.81$$

Answer:

\(s = T(p) = -0.43p + 277.81\)