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Chrome File Edit View History Bookmarks People Window Help 90% Thu 10 Dec 13:04:49 L a E M Problem Set #4 C Economics guestion I che b ezto mheducation.com hm.tpx 0.50 points For each sample, do a test for zero correlation a. r +0.47. n 22, a 0.05, two-tailed test b. r -0.37, n 30, 0.10, two-tailed test c. r +0.61, n 6, 0.05, right-tailed test d. r -0.29, n 67, a 0.01, eft-tailed test (a) at hypotheses is being tested for Sample A? a. Ho: p a 0 versus H1: p 0 b. Ho: p 20 versus H1: p

Solution

here there are four samples. for each of the four samples, the test is for zero correlation

the first sample A. r=0.47 n=22 alpha=0.05 two tailed test

the second sample B. r=-0.37 n=30 alpha=0.10 two tailed test

the third sample C. r=0.61 n=6 alpha=0.05   right tailed test

the fourth sample D. r=-0.29 n=67 alpha=0.01 left tailed test

a) for sample A the test is for two tailed test

so the correct option is a because in b and c the alternative hypothesis are one sided

b) to perform the tests the test statis is given as

T=r*sqrt(n-2)/sqrt(1-r²) which under the null hypothesis follows a t distribution with df n-2

for two sided test H0 is rejected iff |t|>t_alpha/2;n-2

for left sided test H0 is rejected iff t<-t_alpha;n-2

for right sided test H0 is rejected iff t>t_alpha;n-2

where t is the observed value of T and t_alpha;n-2 is the upper alpha point of a t distribution with df n-2

for sample A.

calculated value of test statistic is t=0.47*sqrt(22-2)/sqrt(1-0.47²)=2.381

alpha=0.05 and it is two sided test. so critical value=t_{0.05/2;n-2=20} =upper 0.025 point of a t distribution with df 20=2.086

as it is a both sided test and |t|>t_{0.05/2;n-2=20}    hence H0 is rejected.

for sample B.

calculated value of test statistic is t=-0.37*sqrt(30-2)/sqrt(1-(-0.37)²)=-2.107 so |t|=2.107

alpha=0.10 and it is two sided test. so critical value=t_{0.10/2;n-2=28} =upper 0.05 point of a t distribution with df 28=1.701

as it is a both sided test and |t|>t_{0.10/2;n-2=28}    hence H0 is rejected.

for sample C.

calculated value of test statistic is t=0.61*sqrt(6-2)/sqrt(1-(0.61)²)=1.540

alpha=0.05 and it is right sided test. so critical value=t_0.05;n-2=4 =upper 0.05 point of a t distribution with df 4=2.132

as it is a right sided test and t<t_{0.10/2;n-2=28}    hence H0 is accepted.

for sample D.

calculated value of test statistic is t=-0.29*sqrt(67-2)/sqrt(1-(-0.29)²)=-2.443

alpha=0.01 and it is left sided test. so critical value=-t_0.01;n-2=65 =upper 0.01 point of a t distribution with df 65=-2.385

as it is a left sided test and t<-t_0.01;n-2=65    hence H0 is rejected.

sence summarizing

sample           t_calc           t_crit          decision

a                   2.381        +-2.086        reject

b                  -2.107        +-1.701        reject

c                   1.540          2.132          fail to reject

d                   -2.443        -2.385           reject