Question

the cost of a flight is related to the distance raveled. thus the farther you fly, the more you pay for the flight. use the data to answer the questions, keep in mind that x is the distance in miles and the cost in dollars is y. Miles: 225, 800, 1375, 1675, 1950, 2250. Cost ($) 52.90, 95, 155, 174, 204, 210.

Part A: Find the line of best fit (regression line) and round your answer to the nearest hundredths.

Part B: using the equation you found above (rounded regression line) the cost of a flight that travels 525 is how many dollars (to the nearest dollar)

Answer 1

To find the line of best fit (linear regression), we can use the provided data points:

**Miles (x):** 225, 800, 1375, 1675, 1950, 2250

**Cost (y):** 52.90, 95, 155, 174, 204, 210

### Part A: Finding the Line of Best Fit

To calculate the line of best fit, we can use the formula for the linear regression line, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. **Calculate the means of x and y:**
\[
\bar{x} = \frac{\sum x}{n} = \frac{225 + 800 + 1375 + 1675 + 1950 + 2250}{6} = \frac{8250}{6} = 1375
\]
\[
\bar{y} = \frac{\sum y}{n} = \frac{52.90 + 95 + 155 + 174 + 204 + 210}{6} = \frac{790.90}{6} \approx 131.82
\]

2. **Calculate the slope (m):**
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]
- First, compute \( (x_i - \bar{x}) \) and \( (y_i - \bar{y}) \) for each \( (x, y) \) pair, then multiply those values.
- For the \( x^2 \) values, calculate \( (x_i - \bar{x})^2 \), and then sum them.

3. **Complete calculations:**

Let’s compute it step by step:

| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \((x_i - \bar{x})(y_i - \bar{y})\) | \((x_i - \bar{x})^2\) |
|---------|---------|-------------------|-------------------|------------------------------------|------------------------|
| 225 | 52.90 | 225 - 1375 = -1150 | 52.90 - 131.82 = -78.92 | -1150 * -78.92 = 90758.00 | 1322500 |
| 800 | 95 | 800 - 1375 = -575 | 95 - 131.82 = -36.82 | -575 * -36.82 = 21120.50 | 330625 |
| 1375 | 155 | 1375 - 1375 = 0 | 155 - 131.82 = 23.18 | 0 * 23.18 = 0.00 | 0 |
| 1675 | 174 | 1675 - 1375 = 300 | 174 - 131.82 = 42.18 | 300 * 42.18 = 12654.00 | 90000 |
| 1950 | 204 | 1950 - 1375 = 575 | 204 - 131.82 = 72.18 | 575 * 72.18 = 41461.50 | 330625 |
| 2250 | 210 | 2250 - 1375 = 875 | 210 - 131.82 = 78.18 | 875 * 78.18 = 68317.50 | 765625 |

Now, summing each column:

- \( \sum{(x_i - \bar{x})(y_i - \bar{y})} = 90758 + 21120.50 + 0 + 12654 + 41461.50 + 68317.50 = 184011.50 \)
- \( \sum{(x_i - \bar{x})^2} = 1322500 + 330625 + 0 + 90000 + 330625 + 765625 = 2212875 \)

Now we can compute \( m \):

\[
m = \frac{184011.50}{2212875} \approx 0.0831
\]

Now, we find the intercept \( b \):

\[
b = \bar{y} - m \bar{x} = 131.82 - 0.0831 \cdot 1375 \approx 131.82 - 114.26 \approx 17.56
\]

Thus, the line of best fit is:

\[
y = 0.08x + 17.56 \quad (\text{rounded to the nearest hundredths})
\]

### Part B: Cost of a Flight for 525 Miles

Now using the equation \( y = 0.08x + 17.56 \):

\[
y = 0.08(525) + 17.56
\]
\[
y = 42 + 17.56 \approx 59.56
\]

Rounding to the nearest dollar gives:

\[
\text{Cost} \approx 60
\]

**Final Results:**
- **Part A:** The line of best fit is \( y = 0.08x + 17.56 \).
- **Part B:** The cost of a flight that travels 525 miles is approximately **$60**.