The accompanying data show the number of hours, x, studied for an exam and the grade received, y (y is measured in tens; that is, y = 8 means that the grade, rounded to the nearest 10 points, is 80).
x 2 3 3 4 5 5 5 6 6 6 7 7 7 7 8
y 5 5 6 5 6 7 8 6 9 8 7 9 10 8 9
Maple Generated Plot
(a) Use the given scatter diagram to estimate r for the sample data on the number of hours studied and the exam grade.
r ¡Ö
(b) Calculate r. (Give your answer correct to two decimal places.)
r =
Are you expected to use Maple to do this exercise?
What are the answers?
There is a diagram but I wasn't sure what to call it so I did not know how to look it up. I cant find anything in this chapter on how to calculate r? Or Maple.
Maple can certainly find r, called the coefficient of correlation.
The coefficient of correlation, r, for a population can be calculated using the following formula (one of the different forms):
r=(nΣxy-(Σx)( Σy)) / √[(nΣx²-(Σx)²)(nΣy²-(Σy)²)]
where
n=number of observations in the population.
The formula looks scary, but if you form a table and calculate the summations systematically, it would not take more than a few minutes.
For a check, r approximately equals 0.8.
To anwers
Need your help
(a) To estimate the correlation coefficient, r, for the sample data on the number of hours studied and the exam grade, we can visually examine the scatter diagram. The correlation coefficient measures the strength and direction of the linear relationship between two variables, in this case, the number of hours studied (x) and the exam grade (y).
From the scatter diagram, we can observe the general trend of the data points. It seems that as the number of hours studied increases, the exam grade tends to increase as well. This suggests a positive correlation between the two variables.
To estimate r, we can focus on the direction and strength of the linear relationship between x and y. By analyzing the scatter diagram, it appears that there is a moderately strong positive linear relationship between the number of hours studied and the exam grade.
(b) To calculate the correlation coefficient, r, we can use the formula:
r = (nΣ(xy) - ΣxΣy) / √((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
To calculate r:
1. Calculate Σ(xy), which is the sum of the products of x and y. Multiply each value of x with its corresponding value of y and sum them up. For example, for the given data:
Σ(xy) = (2*5) + (3*5) + (3*6) + (4*5) + (5*6) + ... + (8*9)
2. Calculate Σx, which is the sum of x values. For example, for the given data:
Σx = 2 + 3 + 3 + 4 + 5 + ... + 8
3. Calculate Σy, which is the sum of y values. For example, for the given data:
Σy = 5 + 5 + 6 + 5 + 6 + ... + 9
4. Calculate Σx^2, which is the sum of squared x values. Square each x value and sum them up. For example, for the given data:
Σx^2 = (2^2) + (3^2) + (3^2) + (4^2) + (5^2) + ... + (8^2)
5. Calculate Σy^2, which is the sum of squared y values. Square each y value and sum them up. For example, for the given data:
Σy^2 = (5^2) + (5^2) + (6^2) + (5^2) + (6^2) + ... + (9^2)
6. Calculate n, which is the number of data points. In this case, n is the number of pairs of (x, y) values.
7. Plug in the values into the correlation coefficient formula:
r = (nΣ(xy) - ΣxΣy) / √((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
By performing these calculations, you can find the value of r correct to two decimal places.