Math  /  Data & Statistics

QuestionGiven a sample with r=0.823,n=10r=0.823, n=10, and α=0.05\alpha=0.05, determine the critical values t0t_{0} necessary to test the claim ρ=0\rho=0. \qquad A. ±1.833\pm 1.833 B. ±2.821\pm 2.821 C. ±1.383\pm 1.383 D. ±2.306\pm 2.306

Studdy Solution

STEP 1

1. The sample correlation coefficient is r=0.823 r = 0.823 .
2. The sample size is n=10 n = 10 .
3. The significance level is α=0.05 \alpha = 0.05 .
4. We are testing the null hypothesis ρ=0 \rho = 0 using a two-tailed test.

STEP 2

1. Determine the degrees of freedom.
2. Find the critical t-value for the given significance level and degrees of freedom.
3. Compare the calculated t-value with the provided options.

STEP 3

Calculate the degrees of freedom for the t-distribution. The degrees of freedom df df is given by:
df=n2 df = n - 2
Substitute the given value of n n :
df=102=8 df = 10 - 2 = 8

STEP 4

Find the critical t-value for a two-tailed test with α=0.05 \alpha = 0.05 and df=8 df = 8 .
For a two-tailed test, divide the significance level by 2:
α/2=0.025 \alpha/2 = 0.025
Using a t-distribution table or calculator, find the critical t-value for df=8 df = 8 and α/2=0.025 \alpha/2 = 0.025 .
The critical t-value is approximately ±2.306 \pm 2.306 .

STEP 5

Compare the calculated critical t-value with the provided options:
A. ±1.833 \pm 1.833
B. ±2.821 \pm 2.821
C. ±1.383 \pm 1.383
D. ±2.306 \pm 2.306
The correct answer is option D: ±2.306 \pm 2.306 .
The critical values are:
±2.306 \boxed{\pm 2.306}

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