To determine the correlation of the data set, we can compute the correlation coefficient ($r$), which measures the strength and direction of the linear relationship between the speed (independent variable) and mileage (dependent variable).

Steps to calculate correlation:

1. List the data points:

   Speed (mph)	30	40	50	60	70
   Mileage (mpg)	24	28	30	28	24

2. Formula for the correlation coefficient ($r$):

   [ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} ]

   Where:

   • $x$ is the speed (independent variable),
   • $y$ is the mileage (dependent variable),
   • $n$ is the number of data points (which is 5 here).

3. Calculate necessary sums:

   • $\sum x$ = sum of the speeds,
   • $\sum y$ = sum of the mileages,
   • $\sum x^2$ = sum of the squares of the speeds,
   • $\sum y^2$ = sum of the squares of the mileages,
   • $\sum xy$ = sum of the product of speed and mileage.

Let me calculate the correlation coefficient for you. numpy as np

Data: speed (x) and mileage (y)

speed = np.array([30, 40, 50, 60, 70]) mileage = np.array([24, 28, 30, 28, 24])

Calculate the necessary sums

n = len(speed) sum_x = np.sum(speed) sum_y = np.sum(mileage) sum_x2 = np.sum(speed2) sum_y2 = np.sum(mileage2) sum_xy = np.sum(speed * mileage)

Calculate the correlation coefficient (r)

r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2)) rThe correlation coefficient ($r$) for the data set is 0.0, indicating that there is no linear relationship between speed and mileage in this case.

This result suggests that the relationship between speed and mileage is either nonlinear or not significant enough to be captured by a simple linear model.

Would you like to explore why this happens or discuss the possible trends in the data? Here are a few follow-up questions to consider:

1. How would a nonlinear relationship between speed and mileage affect the correlation?
2. What are other ways to measure the relationship between two variables besides the correlation coefficient?
3. Can the U-shape pattern in this data indicate any specific real-world phenomenon related to vehicle performance?
4. How can I analyze this data further to understand the relationship between speed and mileage?
5. What might be the effect of extreme values or outliers on the correlation coefficient?

Tip: The correlation coefficient only measures linear relationships; a non-linear trend might be invisible to it.