Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find yhe critical values of r given the number of pairs of data n and the significance level a.
N=40, a=0.01
A) r= 0.43
B) r= 0.402
C) r=+-0.402
D) r=+-0.312
Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance level a.
N=17, a=0.05
A) r=0.482
B) r=+-0.482
C) r=+-0.606
D) r=0.497
To find the critical values of r given the number of pairs of data, n, and the significance level, a, we can use a table or formula.
In this case, n=40 and a=0.01. Since the sample size is n=40, we will use the critical value for a two-tailed test at the significance level of 0.01, which corresponds to 0.005 in each tail.
Looking up the critical value in a table for the two-tailed test at 0.01 level, the value is 0.325.
Therefore, the correct answer is option D) r=+-0.312.
The critical values of r can be found using a t-distribution with n-2 degrees of freedom, where n is the number of pairs of data. The formula is:
r_critical = ± t_critical * (1 - r^2)^(1/2) / (n - 2)^(1/2)
where t_critical is the t-value that corresponds to the significance level a and n-2 degrees of freedom. For a=0.01 and n=40, the t_critical value is 2.704.
Plugging in the values:
r_critical = ± 2.704 * (1 - r^2)^(1/2) / 38^(1/2)
We want to solve for r when r_critical = 0.402 (since we're looking for the absolute value of r_critical):
0.402 = ± 2.704 * (1 - r^2)^(1/2) / 38^(1/2)
Squaring both sides:
0.161604 = 7.331296 * (1 - r^2) / 38
1 - r^2 = 0.035133891
r = ± 0.1875059
So the answer is (D) r = ± 0.312 (rounded to three decimal places).
The critical values of r can be found using a t-distribution with n-2 degrees of freedom, where n is the number of pairs of data. The formula is:
r_critical = ± t_critical * (1 - r^2)^(1/2) / (n - 2)^(1/2)
where t_critical is the t-value that corresponds to the significance level a and n-2 degrees of freedom. For a=0.05 and n=17, the t_critical value is 2.109.
Plugging in the values:
r_critical = ± 2.109 * (1 - r^2)^(1/2) / 15^(1/2)
We want to solve for r when r_critical = 0.482 (since we're looking for the absolute value of r_critical):
0.482 = ± 2.109 * (1 - r^2)^(1/2) / 15^(1/2)
Squaring both sides:
0.232324 = 4.452681 * (1 - r^2) / 15
1 - r^2 = 0.491689
r = ± 0.701015
So the answer is (C) r = ± 0.606 (rounded to three decimal places).