Question

Using a population proportion with a confidence interval. A Wholesale store looked into the state of South Carolina and North Carolina to see how many apple tablets were purchased out of a total amount of tablets within the past week. In South Carolina, a sample showed that out of 302 tablets 113 of them were Apple. In North Carolina, a sample showed that out of 340 tablets 195 of them were Apple. Develop and calculate a 95% confidence interval for both the population proportions.

Answer 1

Solution :

Given,

n1 = 302 ....... Sample size

x1 = 113 .......no. of successes in the sample

n2 = 340 ....... Sample size

x2 = 195 .......no. of successes in the sample

Let denotes the sample proportion.

    1 = x1/n1   = 113/302 = 0.374  

  2 = x2/n2   = 195/340 = 0.574  

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 (1 -  1)/n1 +   2 (1 -  2)/n2

= 1.96 * [ 0.374 *(1 - 0.374)/302] + [ 0.574 *(1 - 0.574)/340]

= 0.076

Point estimate = 1 - 2

  = 0.374 - 0.574

= -0.2

Now the confidence interval is given by

( - E)   ( + E)

( -0.2 - 0.076 )   ( -0.2 + 0.076 )

-0.276   -0.124

Required 95% Confidence Interval is ( -0.276 , -0.124 )