Ask a New Question
Menu

The sizes and sale prices for six homes are given in the table. Which correlation coefficient is the only realistic value for the correlation between the sizes of these homes and their sale prices?

Home Sizes and Sale Prices
Size of Home, X (in square feet)
1,200
1,350
1,720
1,510
1,440
1,675
Selling Price, y (in thousands of dollars)
223
298
427
375
310
402

5 answers

To determine a realistic value for the correlation coefficient between the sizes of the homes and their sale prices, we can analyze the relationship using the given data. The correlation coefficient, denoted as \( r \), can range from -1 to +1.

  1. Positive Correlation: If larger homes tend to sell for higher prices, we expect a positive \( r \) close to +1.
  2. No Correlation: If there is no apparent relationship, \( r \) would be around 0.
  3. Negative Correlation: If larger homes tended to sell for lower prices, \( r \) would be a negative value close to -1.

Given the sizes and sale prices of the homes in the data:

  • Sizes (X): 1200, 1350, 1720, 1510, 1440, 1675
  • Prices (Y in thousands): 223, 298, 427, 375, 310, 402

By observing the trend, larger homes generally have higher selling prices.

Now, to calculate the correlation coefficient \( r \):

  1. Calculate the means of X and Y:

    • Mean of sizes: \( \bar{X} = \frac{1200 + 1350 + 1720 + 1510 + 1440 + 1675}{6} \)
    • Mean of prices: \( \bar{Y} = \frac{223 + 298 + 427 + 375 + 310 + 402}{6} \)

  2. Calculate the covariance and the standard deviations of X and Y.

  3. Compute the correlation coefficient using the formula: \[ r = \frac{\text{Cov}(X, Y)}{SD(X) \times SD(Y)} \]

To simplify our answer based on the data provided, you could also use a calculator or statistical software to quickly find \( r \).

Assuming that calculations yield \( r \approx 0.9 \), for instance, it would indicate a strong positive correlation.

In conclusion, considering the data and the positive trend observed, a realistic value for the correlation would be close to +1, indicating a strong positive correlation between home size and selling price. Hence, values like \( r = 0.8, 0.9 \) or similar are realistic, while values like negative \( r \) or those closer to zero would not represent the trend accurately.

Is it
p = 0.981
p = 0.318
p= -0.901
p = -0.018

Given the values you've provided for the correlation coefficient \( p \):

  • \( p = 0.981 \)
  • \( p = 0.318 \)
  • \( p = -0.901 \)
  • \( p = -0.018 \)

Considering the data you provided for home sizes and sale prices, we can analyze each of these options based on the expected relationship:

  1. \( p = 0.981 \): This indicates a very strong positive correlation. Given that larger home sizes typically correspond to higher sale prices, this is a realistic value.

  2. \( p = 0.318 \): This indicates a weak positive correlation. While it suggests some relationship, it doesn’t strongly reflect the likely trend observed between size and price based on your data.

  3. \( p = -0.901 \): This indicates a strong negative correlation, which contradicts the expected positive relationship between size and price. Therefore, this is not realistic.

  4. \( p = -0.018 \): This is very close to zero and indicates no correlation, also contradicting the likely positive relationship expected between size and price.

Based on the context of your dataset, the most realistic correlation coefficient would be \( p = 0.981 \), suggesting a very strong positive correlation between the size of the homes and their sale prices.

It was not p= it was r=

Thank you for the clarification! In statistics, both \( p \) and \( r \) are commonly used to represent correlation coefficients, where \( r \) usually signifies the correlation coefficient specifically.

Given your options for the correlation coefficient \( r \):

  1. \( r = 0.981 \)
  2. \( r = 0.318 \)
  3. \( r = -0.901 \)
  4. \( r = -0.018 \)

As previously mentioned, the most realistic value based on the data you provided (sizes of homes and their selling prices) would be \( r = 0.981 \).

This indicates a very strong positive correlation between the sizes of the homes and their selling prices, suggesting that larger homes tend to sell for higher prices, consistent with typical real estate trends.

The other values indicate either a weak correlation or a negative correlation, which would not accurately reflect the likely relationship observed in your dataset.