The data below represent information on house size (in square metres) and asking price (in thousands of dollars) for a small sample of 7 houses:[table]
House Size | 209 | 210 | 213 | 216 | 210 | 203 | 206 Asking Price | 630 | 619 | 659 | 659 | 645 | 610 | 595
[/table]Calculate the correlation coefficient between House Size and Asking Price (2dp).

Question

BlackTom AI · Accepted Answer

We start by restating the data to ensure clarity: the seven pairs are (House Size, Asking Price) = (209,630), (210,619), (213,659), (216,659), (210,645), (203,610), (206,595).

Step 1: Compute the means of each variable.
- Mean of House Size (x̄): sum of sizes = 209+210+213+216+210+203+206 = 1467; n = 7; x̄ = 1467/7 ≈ 209.5714.
- Mean of Asking Price (ȳ): sum of prices = 4417; n = 7; ȳ = 4417/7 = 631.0.

Step 2: Calculate deviations from the means for each observation.
- For each i, compute dx_i = x_i − x̄ and dy_i = y_i − ȳ.
- Example: for the first house, dx_1 = 209 − 209.5714 ≈ −0.5714; dy_1 = 630 − 631 = −1.
- Repeat for all seven observations to obtain the set of (dx_i, dy_i) terms.

Step 3: Compute the cross‑product sum and the sums of squares.
- Cross‑product sum Sxy = Σ(dx_i * dy_i). When computed across all seven pairs, Sxy ≈ 544.0.
- Sxx = Σ(dx_i^2) ≈ 109.728.
- Syy = Σ(dy_i^2) ≈ 3646.0.

Step 4: Use the Pearson correlation formula.
- The correlation coefficient r = Sxy / sqrt(Sxx * Syy).
- Compute the denominator: sqrt(Sxx * Syy) = sqrt(109.728 * 3646) ≈ sqrt(400,068.5) ≈ 632.5.
- Then r ≈ 544.0 / 632.5 ≈ 0.86.

Step 5: Interpret the result in practical terms.
- A positive correlation around 0.86 indicates a strong positive linear relationship between house size and asking price in this sample: as size increases, price tends to increase as well, and the association is fairly strong.

Notes on reasoning and potential pitfalls:
- Verification of means is crucial; an exact mean for ȳ being 631.0 simplifies the deviations, since several dy_i values become clean integers (e.g., −1, −12, 28, 28, 14, −21, −36).
- The numbers above are rounded at intermediate steps; the final r value close to 0.86 remains robust to small rounding differences.
- If someone computes r using software, they should ensure the same orientation of variables (x = house size, y = price) and consistent population vs. sample formula conventions; here we implicitly used the population-style covariance numerator without any Bessel correction because the standard formula for the Pearson correlation uses the sums of squares in the denominator directly.

COMM111_S2 Quiz 1: Exploratory Data Analysis

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