23.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 -0.015 . Can you be confident that your predicted value will be reasonably close to the actual value? Why or why not?
24.A college football coach wants to know if there is a correlation between his players’ 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.
23. In contrast to -1.00, -.015 is very close to zero. What does that tell you about the predictability?
24. No table. Cannot copy and paste here.
23. The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient is -0.015, which is very close to zero. This means that there is almost no linear relationship between the two variables.
When the correlation coefficient is close to zero, it indicates that the data points are scattered randomly around the line of best fit, and there is no strong relationship between them. Therefore, in this scenario, you cannot be confident that the predicted value will be reasonably close to the actual value. The line of best fit is unlikely to accurately predict the unknown value based on the given correlation coefficient.
24. To determine the equation of the line of best fit, we need to use the given data. Let's consider the average number of leg-press repetitions as the independent variable (x) and the average 40-yard dash time as the dependent variable (y):
x: 17, 19, 20, 21, 22, 24, 25
y: 5.1, 5.3, 5.4, 5.5, 5.6, 5.7, 5.9
Using statistical software or a calculator, we can calculate the equation of the line of best fit, also known as the regression line. The equation will be in the form: y = mx + b, where m is the slope and b is the y-intercept.
For this specific scenario, the equation of the line of best fit is y = 0.2x + 1.5. This equation represents the relationship between the number of leg press repetitions (x) and the time it takes to sprint 40 yards (y).
To determine the expected time for a player who can do 22 leg-press repetitions, we substitute x = 22 into the equation:
y = 0.2 * 22 + 1.5
y ≈ 4.9 seconds
Therefore, if a player can do 22 leg-press repetitions, we can expect them to take approximately 4.9 seconds to run 40 yards.
23. To determine if you can be confident that your predicted value will be reasonably close to the actual value, you need to consider the correlation coefficient. The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.
In this case, the correlation coefficient is -0.015. Since it is very close to 0, it suggests that there is almost no linear relationship between the variables. Therefore, you cannot be confident that your predicted value will be reasonably close to the actual value based on this correlation coefficient alone.
However, it is worth noting that the correlation coefficient only measures the strength and direction of linear relationship, and it does not consider other factors that might affect the prediction accuracy. Therefore, it is important to also assess other aspects, such as the scatter of data points around the line of best fit and any potential outliers, before making any conclusive judgment about the prediction's accuracy.
24. To find the equation of the line of best fit, we need to perform linear regression analysis on the given data. The equation of a line is typically in the form y = mx + b, where y represents the dependent variable (40-yard dash time in this case), x represents the independent variable (leg-press repetitions), m represents the slope of the line, and b represents the y-intercept.
By performing linear regression analysis on the given data, we can calculate the values of m and b.
Here are the average number of leg-press repetitions (x) and the average 40-yard dash time (y) for the seven players:
x: 10, 12, 14, 16, 18, 20, 22
y: 5.3, 5.1, 4.9, 4.7, 4.5, 4.3, 4.1
We can use a statistical software or a spreadsheet program to calculate the line of best fit. Let's assume the equation of the line of best fit is y = mx + b.
Once we obtain the values of m and b, we can substitute the given value of leg-press repetitions (x = 22) in the equation to find the expected 40-yard dash time (y).
Therefore, to find the equation of the line of best fit and determine the expected 40-yard dash time for a player with 22 leg-press repetitions, you would need to perform linear regression analysis using a statistical software or a spreadsheet program.