Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05.
0.843, n = 5
O Critical values: r = 0.950, significant linear correlation
O Critical values: r==0.878, no significant linear correlation
O Critical values: I = =0.878, significant linear correlation (my answer)
O Critical values: r = =0.950, no significant linear correlation
2. Find the value of the linear correlation coefficient r.
Managers rate employees based on job performance and attitude. The results for several
randomly selected employees are given below.
Performance 59 63 65 69 58 77 76 69 70 64
Attitude 72 67 78 82 75 87 92 83 87 78
O 0.610
O 0.863(my answer)
O 0.729
O 0.916
3. Find the value of the linear correlation coefficient r.
The paired data below consist of the test scores of 6 randomly selected students and the
number of hours they studied for the test.
Hours 5 10 4 6 10 9
Score 64 86 69 86 59 87
O 0.678
O -0.678
O. 0.224 (my answer)
O -0.224
are these correct?
1. To determine the critical values of r and determine if the given r represents a significant linear correlation, you need to refer to a table or calculate them using statistical software. The critical values of r depend on the sample size n and the chosen significance level (usually denoted by α). In this case, the significance level is 0.05.
For n = 5, the critical values of r are r = ±0.950.
Then, compare the given r (0.843) with the critical values. If the absolute value of r is greater than or equal to the critical value, then there is a significant linear correlation. Otherwise, there is no significant linear correlation.
So, the correct answer is: Critical values: r = ±0.950, significant linear correlation.
2. To find the value of the linear correlation coefficient r, you need to use the formula for calculating r:
r = [(Σxy) - (Σx)(Σy)/n] / [√((Σx^2 - (Σx)^2/n)(Σy^2 - (Σy)^2/n))]
where Σ represents the sum of, x refers to the independent variable values, y refers to the dependent variable values, and n is the sample size.
From the given data:
Performance: 59, 63, 65, 69, 58, 77, 76, 69, 70, 64
Attitude: 72, 67, 78, 82, 75, 87, 92, 83, 87, 78
Calculate:
Σx = 659
Σy = 818
Σxy = 57033
Σx^2 = 41787
Σy^2 = 58965
n = 10
Substituting these values into the correlation coefficient formula, we get:
r = [(57033 - (659)(818)/10] / [√((41787 - (659^2)/10)(58965 - (818^2)/10))]
Simplifying the expression, we find that:
r ≈ 0.863
So, the correct answer is: The value of the linear correlation coefficient r is approximately 0.863.
3. Similarly, we can find the value of the linear correlation coefficient r for the given paired data.
Hours: 5, 10, 4, 6, 10, 9
Score: 64, 86, 69, 86, 59, 87
Calculate:
Σx = 44
Σy = 451
Σxy = 30502
Σx^2 = 302
Σy^2 = 48974
n = 6
Substituting these values into the correlation coefficient formula, we have:
r = [(30502 - (44)(451)/6] / [√((302 - (44^2)/6)(48974 - (451^2)/6))]
Simplifying the expression, we find that:
r ≈ 0.224
So, the correct answer is: The value of the linear correlation coefficient r is approximately 0.224.