Question

A study examined the interactions in a​ children's museum.​ (Interactions by visitors to the museum included​ show-and-tell, learning,​ teaching, refocusing, participatory​ play, advocating, and disciplining​ interactions.) Researchers observed a sample of 143 meaningful​ interactions, of which 87 were led by children and the rest were led by adult caregivers. Form a 95​% confidence interval around the​ proportion. What is the difference between the highest possible value and the lowest possible value? Round your final answer to three decimal places as needed.

Answer 1

Solution:

Given,

n = 87 ....... Sample size

x = 143 .......no. of successes in the sample

Let denotes the sample proportion.

     = x/n   = 87/143 = 0.608  

Our aim is to construct 95% confidence interval.

c = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.025 and 1- /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

= 1.96

Now , the margin of error is given by

E = /2 *

= 1.96 * [0.608 *(1 - 0.608)/143]

= 0.080

Now the confidence interval is given by

( - E)   ( + E)

(0.608 - 0.080)   (0.608 + 0.080)

0.528 0.688

Required 95% Confidence Interval for proportion is (0.528 , 0.688)

Now ,

difference between the highest possible value and the lowest possible value

= 0.688 - 0.528

= 0.160