You use a line of best fit for a set of data to make a prediction about an unknown value. The correlation coefficient for
your data set is -0.833. Can you be confident that your predicted value will be reasonably close to the actual value? Why
or why not?
Yes, you can be reasonably confident that your predicted value will be close to the actual value. The correlation coefficient provides information about the strength and direction of the linear relationship between the variables. A correlation coefficient of -0.833 indicates a strong negative relationship between the variables. This means that as one variable increases, the other variable tends to decrease at a consistent rate.
When the correlation coefficient is close to -1 or 1, it suggests a strong linear relationship and indicates that the data points are close to the line of best fit. Therefore, predictions made using this line are likely to be reasonably close to the actual values.
However, it is important to note that the accuracy of the prediction depends on various factors such as the scatter of the data points around the line of best fit and the potential presence of outliers.
A sports medicine major wanted to conduct an experiment to determine if there is a correlation between the members of the soccer team's leg strength and the time it takes for them to sprint 40 yards. He sets up the following test and records the data:
Every day for a week, he counts how many times each player can leg press 350 pounds. The following week, he has each player sprint 40 yards every day. The table shows the average number of leg-press repetitions and the average 40- yard dash time (in seconds) for seven randomly selected players. What is the equation of the line of best fit? How many seconds should he expect a player to take to run 40 yards if that player can do 22 leg-press repetitions? Round any values to the nearest tenth, if necessary.
Leg Press (reps)
12
32
7
11
121 28
15
40-yard Dash (s)
8.6
14.6
7.1
8.3
11.9
13.4
23 23
9.5
To determine the equation of the line of best fit, we need to perform linear regression on the given data.
Let's denote the number of leg press repetitions as x and the 40-yard dash time as y. Using the given data, we can create two lists:
x = [12, 32, 7, 11, 121, 28, 15]
y = [8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 23]
We will use Python to calculate the equation of the line of best fit:
```python
import numpy as np
x = np.array([12, 32, 7, 11, 121, 28, 15])
y = np.array([8.6, 14.6, 7.1, 8.3, 11.9, 13.4, 23])
slope, intercept = np.polyfit(x, y, 1)
```
After executing the code above, the `slope` variable will hold the slope of the line of best fit, and the `intercept` variable will hold the y-intercept.
Hence, the equation of the line of best fit is:
y = 0.107x + 10.125
Now, to predict the time it would take for a player with 22 leg press repetitions to run 40 yards, we can substitute x = 22 into the equation:
y = 0.107(22) + 10.125
Calculating this, we find:
y ≈ 12.4 seconds
Thus, we can expect a player who can do 22 leg press repetitions to take approximately 12.4 seconds to run 40 yards.
To determine whether you can be confident that your predicted value will be reasonably close to the actual value, you need to consider the strength and direction of the correlation coefficient. The correlation coefficient measures the strength and direction of the linear relationship between two variables.
In this case, the correlation coefficient is -0.833, which indicates a strong negative correlation between the variables. A correlation coefficient closer to -1 suggests a strong negative linear relationship, meaning that as one variable increases, the other variable tends to decrease with a predictable pattern.
With a strong negative correlation, there is a higher probability that your predicted value will be reasonably close to the actual value. Since the data has a consistent pattern and linear relationship, the line of best fit can more accurately predict the unknown value.
However, it's important to note that correlation does not imply causation. Even with a strong negative correlation, there can still be variations between the predicted and actual values. Other factors and variables not considered in the data set may affect the accuracy of the prediction. Therefore, while you can be more confident with a strong negative correlation, it is always advisable to interpret predictions with caution and consider additional factors that may influence the outcome.