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Read the passage and answer the questions.
A movie cinema is considering significantly reducing the price of their popcorn as they believe…
A movie cinema is considering significantly reducing the price of their popcorn as they believe their customers spend more on drinks when they buy popcorn. They recorded the following data of the daily revenue from popcorn, ‘x’, and the daily revenue from drinks, ‘y’ over 8 randomly selected days:
| Popcorn revenue (‘x’) | Drinks revenue (‘y’) |
|---|---|
| 14 | 22 |
| 12 | 23 |
| 12 | 17 |
| 14 | 24 |
| 16 | 18 |
| 10 | 25 |
| 13 | 23 |
| 12 | 24 |
(a)[1.3333333333333333]
Find $\bar{x}, \bar{y}$.
(b)[1.3333333333333333]
Using $\bar{x}, \bar{y}$, find regression coefficient of y on x.
(c)[1.3333333333333333]
The equation of the regression line y on x is in the form $y = a + bx$. Calculate the values of $a$ and $b$.
Answer
Answer (a)
AIFrom the table as printed: $\sum x = 103$, $\sum y = 176$, so $\bar{x} = \frac{103}{8} = 12.875$ and $\bar{y} = \frac{176}{8} = 22$.
(The printed key uses $13$ for the third value of $x$, which gives $\sum x = 104$, $\bar{x} = 13$, $\bar{y} = 22$.)
Final answer: $\bar{x} = 12.875,\ \bar{y} = 22$
Answer (b)
AI$\sum(x-\bar{x})^2 = \frac{183}{8} = 22.875$ and $\sum(x-\bar{x})(y-\bar{y}) = -17$, so $b_{yx} = \frac{-17}{22.875} = -\frac{136}{183} \approx -0.743$.
(With the key's value $13$ for the third $x$, $b_{yx} = \frac{-22}{22} = -1$.)
Final answer: $-\frac{136}{183} \approx -0.743$
Answer (c)
AI$y - \bar{y} = b_{yx}(x - \bar{x})$: $y - 22 = -\frac{136}{183}\left(x - \frac{103}{8}\right)$, so $y = \frac{5777}{183} - \frac{136}{183}x$.
Hence $a = \frac{5777}{183} \approx 31.57$ and $b = -\frac{136}{183} \approx -0.743$.
(With the key's data, the line is $y = 35 - x$: $a = 35$, $b = -1$.)
Final answer: $a = \frac{5777}{183} \approx 31.57,\ b = -\frac{136}{183} \approx -0.743$
From ISC 2025 Practice Mathematics, question 120.
Check your working with the Regression lines calculator: both regression lines, b_yx, b_xy and r from data, summary sums or the two lines; with estimates.