Question

o 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. Tost the daim 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) 29.4 242 25.0 26.6 18.7 Per capita personal Income ly) 33,746 30,057 25,156 29,372 28,901 Click the loon to view a table of critical values of Spearman's rank correlation coefficient OA. Hyp against Hp O Oc. Hyp against H.: p0 OE. Hp against HIPO B. Hp against > OD. Hyp> against PHO OF H P O against HIPO Determine the test statistic. .-05 (Round to three decimal place as needed) Specify the rejection region Round to three decimal place as needed.) A>

Answer 1

We have to test the claim that regions with higher percentage of the population with atleast a bachelor's degree have higher per capita personal income at 0.05 level of significance.

1) -

Hypothesis -

Null hypothesis - H₀ : There is no relationship between bachelor's degree & per capitta personal income. i.e. $\rho$ = 0.

Alternative hypothesis - H₀: There is relationship between bachelor's degree & per capitta personal income. i.e. $\rho$ $\neq$ 0.

2) -

Test statistic -

$r_{s} = \frac{\sum (x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum (x_{i}-\bar{x})^{2}\sum (y_{i}-\bar{y})^{2} }}$

Observation table -

Sr. No.	x	y	(x-x_bar)	(y-y_bar)	(x-x_bar)(y-y_bar)	(x-x_bar)^2	(y-y_bar)^2
1	28.4	33746	3.82	4299.2	16422.944	14.5924	18483120.64
2	24.2	30057	-0.38	610.2	-231.876	0.1444	372344.04
3	25	25158	0.42	-4288.8	-1801.296	0.1764	18393805.44
4	26.6	29372	2.02	-74.8	-151.096	4.0804	5595.04
5	18.7	28901	-5.88	-545.8	3209.304	34.5744	297897.64
Total	122.9	147234			17447.98	53.568	37552762.8

Calculations -

$r_{s} = \frac{\sum (x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum (x_{i}-\bar{x})^{2}\sum (y_{i}-\bar{y})^{2} }}$

$r_{s} = \frac{17447.98}{\sqrt{(53.568)(37552762.8) }}$

T's 17447.98 44851.1583

Test statistic -

3) -

Rejection region -

$r_{s}$ > 0.879

a normal distribution N 0.01 0.05 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 0.8687 0.8234 0.8240 0.8351 0.8474 0.8590 0.8689 0.8765 0.8838 0.8918 0.8974 0.9029 0.9080 0.9121 0.9160 0.9196 0.9230 0.9256 0.9285 0.9308 0.9334 0.9356 0.9370 0.9393 0.9413 0.9428 0.9441 0.9462 0.9476 0.9490 0.9505 0.9521 0.9530 0.9540 0.9551 0.9555 0.9568 0.9576 0.9589 0.9593 0.9609 0.8790 0.8666 0.8786 0.8880 0.8970 0.9043 0.9115 0.9173 0.9223 0.9267 0.9310 0.9343 0.9376 0.9405 0.9433 0.9452 0.9479 0.9498 0.9515 0.9535 0.9548 0.9564 0.9575 0.9590 0.9600 0.9615 0.9622 0.9634 0.9644 0.9652 0.9661 0.9671 0.9678 0.9686 0.9693 0.9700 0.9704 0.9712 0.9719 0.9723 0.9730 40 41 42 43