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QUESTION 2 The marks out of 100, obtained by 10 learners in Mathematics and Music were recorded as follows: ... Use DIAGRAM SHEET (page) and draw a scatter plot of the data

Mathematics

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Answers

3 parts
2.2

y=82.939−0.584xy = 82.939 - 0.584x

2.3
-0.544
2.5
(90, 89)

Step-by-step solution

6 steps

Part 2.1

To draw a scatter plot

  • Draw a Cartesian plane with the horizontal axis representing "Mathematics Marks" and the vertical axis representing "Music Marks".
  • Label both axes from 0 to 100 with appropriate increments.
  • Plot each pair of marks $(x, y)$ as a point on the graph. For example, the first point would be $(34, 71)$.

Part 2.2

y=82.939−0.584xy = 82.939 - 0.584x

To determine the equation of the regression line y=a+bxy = a + bx, we first calculate the necessary sums from the given data

n=10n = 10
∑x=540\sum x = 540
∑y=514\sum y = 514
∑x2=35704\sum x^2 = 35704
∑y2=33972\sum y^2 = 33972
∑xy=23934\sum xy = 23934

  1. Calculate the slope bb.

    b=n∑xy−(∑x)(∑y)n∑x2−(∑x)2b = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2} b=10(23934)−(540)(514)10(35704)−(540)2b = \frac{10(23934) - (540)(514)}{10(35704) - (540)^2} b=239340−277560357040−291600b = \frac{239340 - 277560}{357040 - 291600} b=−3822065440≈−0.584b = \frac{-38220}{65440} \approx -0.584

  2. Calculate the y-intercept aa.

    First, find the means: xˉ=54010=54\bar{x} = \frac{540}{10} = 54 and yˉ=51410=51.4\bar{y} = \frac{514}{10} = 51.4.

    a=yˉ−bxˉa = \bar{y} - b\bar{x} a=51.4−(−0.584046...)(54)a = 51.4 - (-0.584046...)(54) a=51.4+31.5385...≈82.939a = 51.4 + 31.5385... \approx 82.939

  3. Write the equation of the regression line.

    y=82.939−0.584xy = 82.939 - 0.584x

    The equation of the regression line is y=82.939−0.584x\boxed{y = 82.939 - 0.584x}.

Part 2.3-0.544

To determine the correlation coefficient rr

r=n∑xy−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}

  1. Use the previously calculated numerator and the first part of the denominator.

    Numerator: n∑xy−(∑x)(∑y)=−38220n \sum xy - (\sum x)(\sum y) = -38220
    Denominator part 1: n∑x2−(∑x)2=65440n \sum x^2 - (\sum x)^2 = 65440

  2. Calculate the second part of the denominator.

    n∑y2−(∑y)2=10(33972)−(514)2n \sum y^2 - (\sum y)^2 = 10(33972) - (514)^2 =339720−264196=75524= 339720 - 264196 = 75524

  3. Calculate rr.

    r=−38220(65440)(75524)r = \frac{-38220}{\sqrt{(65440)(75524)}} r=−382204939769760r = \frac{-38220}{\sqrt{4939769760}} r=−3822070283.509...≈−0.544r = \frac{-38220}{70283.509...} \approx -0.544

    The correlation coefficient is −0.544\boxed{-0.544}.

    2.4 The correlation coefficient r=−0.544r = -0.544 indicates a moderate negative linear correlation between Mathematics marks and Music marks. This means that as a learner's Mathematics mark increases, their Music mark tends to decrease, but the relationship is not very strong.

Part 2.5(90, 89)

An outlier is a data point that significantly deviates from the general trend of the other data points.

Looking at the data, most points show a negative relationship (high in one subject, low in the other). However, the point (90,89)(90, 89) represents a learner who scored high in both Mathematics (90) and Music (89), which goes against the observed negative trend.
The outlier in the set of data is (90,89)\boxed{(90, 89)}.

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