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.

类似问题

Of the following, the correlation coefficient that indicates the weakest relationship between the two variables being measured is ______________.

The correlation in between mobile usage and exam score of a person found to be -2.2. What is your inference from the above statement.

The correlation between temperature in ◦F and age in weeks was r = −0.70. If we converted the temperature to ◦C and age to months, what would the 29 correlation be?

Their correlation coefficients will be the same, since the correlation coefficient is unitless.

r = −0.85

r = −0.03

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单