Construct the indicated confidence interval for the difference in proportions. Assume that the samples are independent and that they have been randomly selected. A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. Construct a 99% confidence interval for the difference in the proportions of New Yorkers and Californians who knew the product.

CI 99%   =  ( - 0.0009 ;  0.0541 )

Step-by-step explanation:

Sample 1 New Yorkers

sample size    n₁   =  558

x₁  =  193

p₁ = x₁/n₁   =  193/ 558      p₁ =  0.3458  q₁ =  1  - p₁   q₁  =  1  -  0.3458

q₁  =  0.6542

Sample 2  Californians

sample size     n₂  =  614

x₂  =  196

p₂  =  x₂/n₂  =  196 / 614    p₂ = 0.3192   q₂  =  1  -  p₂   q₂ =  1 - 0.3192

q₂ =  0.6808

CI 99 %    means   significance level α  = 1 αα%    α = 0.01

α/2  =  0.005

In z-table we look for z score for  0.005    z (c) =  2.57

CI 99 %  =  [  (  p₁  -  p₂ ) ±  z(c) * √( p₁*q₁)/n₁  + ( p₂*q₂)/n₂

p₁ - p₂  =  ( 0.3458 - 0.3192 )  =  0.0266

√( p₁*q₁)/n₁  + ( p₂*q₂)/n₂   =√  0.3458*0.6542)/558 +  0.3192*0.6808)/614

√( p₁*q₁)/n₁  + ( p₂*q₂)/n₂   = √ 4.05*10⁻⁴   +  3.54 * 10⁻⁴

√( p₁*q₁)/n₁  + ( p₂*q₂)/n₂   =  10⁻² * √7.59    =  10⁻²  *  2.75

Then:

CI 99 %  =  0.0266  ±  2.75 * 10⁻²

CI 99%   =  0.0266   ±  0.0275

CI 99%   =  ( - 0.0009 ;  0.0541 )

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step-by-step explanation:

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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