Question

Applicants for a job, which involves extensive travel in Spanish speaking countries, must take a proficiency test in Spanish. The sample data below were obtained in a study of the relationship between the numbers of years applicants have studied Spanish (x) and their score on the test (y). Conduct a formal hypothesis test to test for a linear correlation between the number of years a person has studied Spanish and their score on the test. Use a 0.05 significance level.

x	3	4	4	2	5	3	4	5	3	2
y	57	78	72	58	89	63	73	84	75	48

What is the p-value? Round to 4 decimal places.

Answer 1

Let X denote the number of years applicants have studied Spanish

Y denote their score on the test.

Let denote the true correlation coefficient between X and Y.

To test: H₀: That is true correlation coefficient is not significantly different from zero.

versus H₁: That is true correlation coefficient is significantly different from zero.

For the given data, we calculate sample correlation coefficient r.

T

Test statistic:

T = 6.252

T follows t-distribution with n-2 = 8 degrees of freedom.

P-value = P[ |t₈| > 15.15166] = 2*P[ t₈> 6.252] = 0.0002

We reject H₀ at 0.05 level of significance if p-value < 0.05

Hence, we reject H₀ at 0.05 level of significance.

Hence, linear correlation between number of years a person has studied Spanish and their score is significantly different than zero.

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