To solve this problem, we will tackle it step-by-step:

Part (a): Draw a scatter diagram of the data

The data points are given as:

• (2, 4)
• (4, 8)
• (7, 10)
• (7, 7)
• (9, 20)

We need to compare the given graphs with this data set. Let's break down the options:

• Option A: Matches with points (2, 4), (4, 8), (7, 10), (7, 7), but doesn't include (9, 20) correctly.
• Option B: Matches exactly with all points.
• Option C and D: Both show points that don’t match the provided data set (C and D include (2, 20) and other incorrect points).

Thus, Option B is the correct scatter diagram.

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Part (b): Compute the Correlation Coefficient

We will use the formula for Pearson's correlation coefficient, $r$, which is:

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

Where:

• $n$ is the number of data points,
• $x$ and $y$ are the values of the variables.

Step 1: Calculate the necessary sums:

x	y	x*y	x²	y²
2	4	8	4	16
4	8	32	16	64
7	10	70	49	100
7	7	49	49	49
9	20	180	81	400

Now, sum each column:

$\sum x = 2 + 4 + 7 + 7 + 9 = 29$ $\sum y = 4 + 8 + 10 + 7 + 20 = 49$ $\sum xy = 8 + 32 + 70 + 49 + 180 = 339$ $\sum x^2 = 4 + 16 + 49 + 49 + 81 = 199$ $\sum y^2 = 16 + 64 + 100 + 49 + 400 = 629$

Step 2: Apply the values into the formula:

[ r = \frac{5(339) - (29)(49)}{\sqrt{[5(199) - (29)^2][5(629) - (49)^2]}} ]

Simplify the expression:

[ r = \frac{1695 - 1421}{\sqrt{[995 - 841][3145 - 2401]}} ] [ r = \frac{274}{\sqrt{[154][744]}} ] $r = \frac{274}{\sqrt{114576}}$ $r = \frac{274}{338.57} \approx 0.809$

Thus, the correlation coefficient $r$ is approximately 0.809.

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Part (c): Determine if there is a linear relationship

Since the correlation coefficient $r = 0.809$ is close to 1, this indicates a strong positive linear relationship between $x$ and $y$.

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Would you like more details on any part of the solution, or have any questions?

Here are 5 follow-up questions you can explore:

1. What is the interpretation of the correlation coefficient $r$?
2. How can the scatter diagram visually represent correlation strength?
3. What are the limitations of Pearson's correlation?
4. How does changing one data point affect the correlation coefficient?
5. What are alternative methods for determining non-linear relationships?

Tip: Always check if a scatter plot suggests any outliers before interpreting correlation, as they can strongly influence the result.