Question

A study is conducted to investigate the impact of depression on a patient’s ability to survive cardiac disease. They evaluated 477 participants for depression and followed for 4 years. There were 374 patients with no depression (ND) and 69 of them died. Out of 103 patients with minor or major depression (D), 28 died. Construct a 95% confidence interval for the difference in proportion of deaths between depressed patients with cardiac disease and non-depressed patients with cardiac disease. (Hint: Round sample proportions and SE to four decimal places.)

A) (0.0392, 0.1356)

B) (-0.0072, 0.1818)

C) (0.0080, 0.1666)

D) (-0.0371, 0.2118)

E) (-0.0572, 0.2320)

Answer 1

Solution:

Given:

depressed patients with cardiac disease:

n1 = 103

x1=28

non-depressed patients with cardiac disease.

n2 = 374

x2 = 69

We have to find  a 95% confidence interval for the difference in proportion of deaths between depressed patients with cardiac disease and non-depressed patients with cardiac disease.

Formula:

(pi - p2 - E, pi-p2 + E)

where

P1 = în _ 11 28 = 0.2718 = 103

6-29 2 = = 374 = 0.1845

E = 2 x S Epipi

SEpi-Pa = Px (1 –p) – P2 x (1 – 12) ni n2

0.2718 x (1 – 0.2718) SEpi-p2 = V 0.1845 x (1 – 0.1845) 374 103

SEpi-p = 0.00192180 + 0.00040229

SEpi-p = 0.00232408

SEpi-p = 0.0482

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

0.1 V123 .00 .01 .02 .03 .04 .05 .06 0.0 ,5000 15040 5080 5120 .5160 5199 39 1.5398 .5438 5478 1.5517 .5557 .5596 .636 0.2 15793 5832 ,5871 1.5910 .5948 1.5987 ,6026 0.3 .6179 .6217 .6255 .6293 .6331 1.6368 | 6406 0.4 .6554 ,6591 ,6628 .6664 .6700 .67366 16772 0.5 ,6915 ,6950 .6985 ,7019 .7054 7088 0.6 72577291 .7324 .7357 7389 7422 F454 0. 7 7580 7611 ,7642 .7673 704 7734 7764 0.8 1.788179107939 .7967 7995 ,8023 4051 09819816,8212 ,82388 .82648289 1.4315 18413 18438 ,8461 ,8485 .8508 18531 .4554 .8643 18665 8686 .8708 ,8729 ,8749 70 18849 1.8869 .888 ,8907 .8925 .894 962 1.3 19032 9049 ,9066 19082 19099 9115 1931 1.4 19192 .92079222 .9236 .9251 ,9265979 1.5 ,9332 19345 .9357 .9370 9382 19394 9606 1.6 19452 19463 19474 .9484 .9495 19505 15 1.7 ,9554 19564 19573 9582 19591 .9599 9508 1.8 19641 .9649 .9656 .9664 .9671 .96789 586 | 19 7 47- 0710 -07- 19750 2.0 ,9772 19778 19783 19788 19793 19798 19803 1.0 1.1 1.2

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

E = 2 x S Epipi

E = 1.96 x 0.0482

E = 0.0945

Thus

(pi - p2 - E, pi-p2 + E)

(0.2718 -0.1845 -0.0945,0.2718 -0.1845 +0.0945)

(-0.0072.0.1818)

Thus correct option is:

B) (-0.0072, 0.1818)