Question

Sam works at the Regional Conservatorium. She wonders how many people in the town are aware that there is a Conservatorium in the town since it is hidden down a driveway off a side street of the CBD. Sam is quite busy, so she employed one of her high-school aged students to undertake a survey of the town’s population of 20,000 people. The student surveyed 110 people over a 3-hour period. Of those surveyed, 78 people indicated that they did indeed know there was a Conservatorium in the town. b. Based upon this sample, determine a 95% confidence interval estimate for the proportion of the population who know about the Conservatorium.

Answer 1

The student surveyed 110 people over a 3-hour period

Of those surveyed, 78 people indicated that they did indeed know there was a Conservatorium in the town

Population of town 20,000

Thus N = 20000

         n = 110

Let x denotes number of succes

Of those 110 people surveyed, 78 people indicated that they did indeed know there was a Conservatorium in the town

Therefore x = 78

To form a sample proportion, take x, the random variable for the number of successes and divide it by n, (the sample size). The random variable p is the sample proportion.

Thus p is given by

        p = x / n   = 78 / 110 = 0.7090909

        p = 0.7090909

Thus p is proportion of people in sample who indeed know there was a Conservatorium in the town.

Let q =1 - p = 0.2909091

To determine a 95% confidence interval estimate for the proportion of the population

thus = 0.05

The confidence interval for a population proportion P , therefore, becomes:

          

P =p [ * ]

Where    =    = 0.7090909 * 0.2909091 / 110

                                    = 0.001875282

                                    = 0.04330453

Also = 1.96            at 5 % level of significance .

P = p [ * ] = 0.7090909 1.96 *0.04330453

       

95% confidence interval C.I. for population proportion P

                      C.I.      = { 0.624214 , 0.7939678 }

Note :

{

Let X be number of people in town ( Population of 20,000) that they did indeed know there was a Conservatorium in the town

So we find Interval for X also

C.I. for population proportion P is { 0.624214 , 0.7939678 }

        Where P = X / N

                     Lower bound = 0.624214 and upper bound = 0.7939678

Thus for N = 20000

X=     ( X / N = P = 0.624214    , X / N = P = 0.7939678 )

X=       ( X = N * 0.624214          ,     X = N *0.7939678 )

X =      ( 12484.28     , 15879.36 )

}