Questions: The table below shows the heights (in feet) and the number of stories of six notable buildings in a city. Complete parts a - f below. Height, x 775 619 519 508 491 474 Stories, y 53 47 46 42 38 35 (e) Predict the value of y for x=321. Choose the correct answer below. A. 47 B. 31 C. 51 D. not meaningful because the x value is outside the scope of the data

Solution Steps

The table provides the heights (in feet) and the number of stories of six notable buildings in a city.

• Heights (x): 775, 619, 519, 508, 491, 474
• Stories (y): 53, 47, 46, 42, 38, 35

To predict the value of y for x = 321, we need to determine the relationship between the height (x) and the number of stories (y). This can be done using linear regression or by observing the trend in the data.

Using the least squares method, we calculate the linear regression equation \( y = mx + b \).

1. Calculate the means of x and y: \[ \bar{x} = \frac{775 + 619 + 519 + 508 + 491 + 474}{6} = 564.33 \] \[ \bar{y} = \frac{53 + 47 + 46 + 42 + 38 + 35}{6} = 43.5 \]

2. Calculate the slope (m): \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \] \[ m = \frac{(775-564.33)(53-43.5) + (619-564.33)(47-43.5) + (519-564.33)(46-43.5) + (508-564.33)(42-43.5) + (491-564.33)(38-43.5) + (474-564.33)(35-43.5)}{(775-564.33)^2 + (619-564.33)^2 + (519-564.33)^2 + (508-564.33)^2 + (491-564.33)^2 + (474-564.33)^2} \] \[ m \approx 0.072 \]

3. Calculate the y-intercept (b): \[ b = \bar{y} - m\bar{x} \] \[ b = 43.5 - 0.072 \times 564.33 \approx 2.99 \]

4. The linear regression equation is: \[ y = 0.072x + 2.99 \]

Substitute \( x = 321 \) into the linear regression equation: \[ y = 0.072 \times 321 + 2.99 \approx 26.11 \]

Final Answer