Question

An instructor at a major research university occasionally teaches summer session and notices that that there are often students repeating the class. Out of curiosity, she designs a random sample of students enrolled in summer sessions and counts the number repeating a class. She counts 105 students in the sample, of which 19 are repeating the class. The university enrolls 15,000 students.

A 95% confidence interval is given by:

(0.09, 0.272).

(0.107, 0.255).

(0.15, 0.23).

(0.345, 0.453).

Answer 1

Solution :

Given,

n = 105 ....... Sample size

x = 19 .......no. of successes in the sample

Let denotes the sample proportion.

     = x/n   = 19 / 105 = 0.181  

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 * [ 0.181 *(1 - 0.181 )/105]

= 0.074

Now the confidence interval is given by

( - E)   ( + E)

( 0.181 - 0.074 )   ( 0.181 + 0.074 )

   0.107   0.255  

Required 95% Confidence Interval is ( 0.107 , 0.255 )