Question

The following data represent the percentage of the population that have at least a bachelor's degree and the per capita personal income for randomly chosen regions. Test the claim that regions with a higher percentage of the population with at least a bachelor's degree have a higher per capita personal income at the -0.05 level of significance Bachelor's degree (x) 24.9 29.6 26.7 24.4 17.6 Per capita personal Income 33.741 29,369 24,671 28,955 29,984 ly) Click the icon to view a table of critical values of Spearman's rank correlation coefficient Identify the null and alternative hypotheses. OA. Hip#0 against He: p=0 O C. Ho: p=0 against Hy:p> O E. Hy: p=0 against Hyp<0 O B. Ho:p>0 against He: p=0 OD. Ho: p<0 against Hap=0 OF Hop against Hopio Determine the test statistic T. (Round to three decimal place as needed.)

Answer 1

( X)	( Y)	X^2	Y^2	X*Y
24.9	33741	620.01	1138455081	840150.9
29.6	29369	876.16	862538161	869322.4
26.7	24671	712.89	608658241	658715.7
24.4	28955	595.36	838392025	706502
17.6	29984	309.76	899040256	527718.4

calculation procedure for correlation
sum of (x) = 123.2
sum of (y) = 146720
sum of (x^2) = 3114.18
sum of (y^2) = 4347083764
sum of (x*y) = 3602409.4
to calculate value of r( x,y) = co-variance ( x,y ) / sd (x) * sd (y)
co-variance ( x,y ) = [ sum (x*y - N *(sum (x/N) * (sum (y/N) ]/n-1
= 3602409.4 - [ 5 * (123.2/5) * (146720/5) ]/5- 1
= -2554.28
and now to calculate r( x,y) = -2554.28/ (SQRT(1/5*3602409.4-(1/5*123.2)^2) ) * ( SQRT(1/5*3602409.4-(1/5*146720)^2)
=-2554.28 / (3.9631*2889.0166)
=-0.2231
value of correlation is =-0.2231

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Given that,
value of r =-0.2231
number (n)=5
null, Ho: row(ρ) =0
alternate, H1: row(ρ)<0
level of significance, alpha = 0.05
from standard normal table,left tailed t alpha/2 =2.353
since our test is left-tailed
reject Ho, if to < -2.353
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=-0.2231/(sqrt( ( 1--0.2231^2 )/(5-2) )
to =-0.396
|to | =0.396
critical value
the value of |t alpha| at los 0.05% is 2.353
we got |to| =0.396 & | t alpha | =2.353
make decision
hence value of |to | < | t alpha | and here we do not reject Ho
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[ANSWERS]
Ho: row(ρ) = 0 H1: row(ρ)<0
test statistic: -0.396
reject Ho, if rs < -2.353
do not reject Ho, there is not sufficient evidence to support the claim