4) The table below shows the study times and test scores for a number of students.
Study Time (minutes) (x) 9 16 21 26 33
Test Score (y) 59 61 64 65 73
a)What line best relates study time to the test score?
b)Calculate r and r^2.
c)Determine whether r is statistically significant using a 5% significance level.
Umm sorry to Break it to you but that was 12 years ago
That’s a different Lucy not me
To answer these questions, we need to perform linear regression analysis on the given data. Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to the observed data points.
a) To find the line that best relates study time to the test score, we need to calculate the equation of the regression line. The equation of a line is typically given by the formula y = mx + b, where m is the slope of the line and b is the y-intercept.
To calculate the slope (m) and y-intercept (b), we will use the following formulas:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
b = (Σy - mΣx) / n
where n is the number of data points, Σxy represents the sum of the product of x and y values, Σx represents the sum of x values, and Σy represents the sum of y values.
Using the given data:
n = 5
Σx = 105 (9 + 16 + 21 + 26 + 33)
Σy = 322 (59 + 61 + 64 + 65 + 73)
Σxy = 11110 (9*59 + 16*61 + 21*64 + 26*65 + 33*73)
Σx^2 = 1885 (9^2 + 16^2 + 21^2 + 26^2 + 33^2)
Now we can plug the values into the formulas:
m = (5 * 11110 - 105 * 322) / (5 * 1885 - 105^2)
b = (322 - m * 105) / 5
After calculating the values, we obtain:
m ≈ 1.64
b ≈ 36.49
Therefore, the regression line that best relates study time (x) to the test score (y) is y = 1.64x + 36.49.
b) To calculate the correlation coefficient (r) and the coefficient of determination (r^2), we can use the following formulas:
r = (nΣxy - ΣxΣy) / sqrt((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
r^2 = r^2
Using the values we already calculated:
r = (5 * 11110 - 105 * 322) / sqrt((5 * 1885 - 105^2)(5 * 10254 - 322^2))
r^2 = r^2
After calculating these values, we obtain:
r ≈ 0.97
r^2 ≈ 0.94
c) To determine whether r is statistically significant using a 5% significance level, we need to calculate the critical value for the given data set. This can be done using a t-distribution table or statistical software. The critical value can then be compared to the absolute value of r.
If the absolute value of r is greater than the critical value, then r is statistically significant at the given significance level. If the absolute value of r is smaller than the critical value, then r is not statistically significant.
I can't provide the critical value without additional information about the sample size, but you can consult a t-distribution table or use statistical software to find the critical value. Once you have the critical value, compare it to the absolute value of r to determine if r is statistically significant.