Question

A student, Samuel, speculates that exam scores may be linearly related to hours spent studying. Samuel collects five data points, (X,Y), where X represents hours studied and Y represents exam score -- Ricardo: (2,45); Janette: (4,80); Calvin: (7,95), Edith: (3,55); Joachim: (1,30). The mean score is 61. Samuel uses the equation Y = 20 X to represent the data. Calculate the R-squared value of this line. Assuming that an R-squared value less than 0.4 is "poor", a value between 0.4 and 0.8 is "fair", and a value above 0.8 is "good", how well does this line Y=20X represent the data?

Answer 1

The R-squared value of the line Y=20X is approximately 0.67, indicating a good linear relationship between hours studied and exam scores.

Step-by-step explanation:

The R-squared value measures the proportion of the total variation in the dependent variable (exam scores) that is explained by the independent variable (hours spent studying).

To calculate the R-squared value, we need to first calculate the sum of squares of the residuals (SSR), which measures the total unexplained variation. Then, we calculate the sum of squares of the total (SST), which measures the total variation. Finally, the R-squared value is obtained by subtracting SSR from SST and dividing by SST.

Using the equation Y = 20X, we can calculate the R-squared value as follows:

1. Calculate the mean exam score: (45+80+95+55+30)/5 = 61
2. Calculate the sum of squares of the residuals (SSR): SSR = (45-40)^2 + (80-60)^2 + (95-80)^2 + (55-60)^2 + (30-20)^2 = 225 + 400 + 225 + 25 + 100 = 975
3. Calculate the sum of squares of the total (SST): SST = (45-61)^2 + (80-61)^2 + (95-61)^2 + (55-61)^2 + (30-61)^2 = 256 + 361 + 1344 + 36 + 961 = 2958
4. Calculate the R-squared value: R-squared = (SST - SSR)/SST = (2958 - 975)/2958 = 1983/2958 ≈ 0.67

Therefore, the R-squared value of the line Y = 20X is approximately 0.67. This value falls within the range of 0.4 and 0.8, which means that the line represents the data fairly well, indicating a good linear relationship between hours studied and exam scores