Question

Assignment for Sections 7.3 - 7.4 1) The mayor of a small city has suggested that the state locate a new prison there, arguing that the construction project and resulting jobs will be good for the economy. A total of 183 residents show up for a public hearing on the proposal, and a show of hands finds only 31 in favor of the prison project. Construct a 94% confidence interval for the true proportion of residents of that city who favor the prison project, and write a sentence that interprets the interval. What can the city council conclude about public support for the mayor's initiative?

Answer 1

Solution:

Given: The mayor of a small city has suggested that the state locate a new prison there, arguing that the construction project and resulting jobs will be good for the economy.

n = sample size = 183

x = Number of people favor of the prison project.

We have to find 95% confidence interval for true proportion of residents of that city favor of the prison project.

Formula:

-E, P+E)

where

31 182 = 0.1694

E = Z X P * (1-P)

Find Area = ( 1 + c ) / 2 = ( 1 + 0.94) /2 = 1.94 / 2 = 0.9700

Look in z table for Area = 0.9700 or its closest area and find z value

2 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 1 1.3 11.4 11.5 11.6 11.7 1 | 18 .00 .01 .02 .03 .04 .05 .5000 5040 5080 5120 5160 .5199 5398 5438 5478 5517 5557 .5596 .5793 5832 .5871 5910 .5948 5987 .6179 6217 .6255 .6293 6331 .6368 6554 6591 6628 6664 6700 .6736 .6915 6950 .6985 7019 7054 .7088 .7257 7291 7324 | 7357 7389 7422 .7580 7611 7642 7673 7704 7734 .7881 7910 7939 7967 .7995 8023 .8159 8186 8212 .8238 .8264 .8289 1.8413 - .8438 8461 .8485 .8508 .8531 .8643 .8665 .8686 .8708 .8729 .8749 .8849 .8869 .8888 .8907 | 8925 1.8944 1.9032 9049 9066 9082 1.9099 19192 .9207 9222 .9236 | 1.9251 | 9265 9332 9345 .9357 93701 | 9382 | 9394 .9452 9463 9474 9484 .9495 9505 9554 .9564 .9573 9582 .9591 .9599 9041 1909 96560664 0624 9070 .06 .07 .08 1 1.09 .5239 5279 519 5359 .5636 5675 5114 | 5753 .6026 6064 6173 | 6141 6406 6443 640 6517 6772 6808 6844 6879 .7123 7157 7190 | 7224 7454 7486 7517 7549 7764 7794 783 7852 .8051 .8078 .81106 8133 .8315 .8340 .8355 8389 .8554 8577 .859 8621 8770 8790 | | 8810 8830 .8962 8980 1,8997 9015 1.9131 9147 19162 .9279 .9292 9305 9406 9418 9429 19441 9515 .9525 .9595 9545 .9608 9616 925 | -90069692-96991 19706 1.9177 -9319 19633

Area 0.9699 closest to 0.9700 and it corresponds to 1.8 and 0.08

thus Z_c = 1.88

Thus

E = Z X P * (1-P)

E= 1.88 x 1 0.1694 x (1 – 0.1694) 183

0.1694 x 0.8306 E = 1.88 Xv 183 V

E = 1.88 X V0.0007689

E = 1.88 x 0.0277285

E = 0.0521295

E = 0.05213

Thus

-E, P+E)

(0.1694 -0.05213.0.1694 +0.05213)

(0.11727,0.22153)

(0.1173,0.2215)

Thus we are 94% confident that the true proportion of residents of that city favor of the prison project is between: 0.1173 to 0.2215 that is between 11.73% to 22.15%.

Since limits of this confidence interval are less than 50%, there is not majority of public support for the mayor's initiative.