A police officer would like to estimate the proportion of petty theft cases that have gone unsolved in a large city. He takes a random sample of 81 petty theft cases and finds 52 of these have gone unsolved. Set up the equation for the 95% confidence interval for p.

95% confidence interval for p is [0.54 , 0.75].

Step-by-step explanation:

We are given that a police officer would like to estimate the proportion of petty theft cases that have gone unsolved in a large city. He takes a random sample of 81 petty theft cases and finds 52 of these have gone unsolved.

Let p = population proportion of petty theft cases

So, the pivotal quantity for 95% confidence interval for population proportion p is given by;

           P.Q. = ~ N(0,1)

where, = proportion of petty theft cases that have gone unsolved in a large city in a sample of 81 cases =

           n = sample of cases = 81

So, 95% confidence interval for the population proportion, p is ;

P(-1.96 < N(0,1) < 1.96) = 0.95

P(-1.96 < < 1.96) = 0.95

P( < < ) = 0.95

P( < p < ) = 0.95

95% confidence interval for p = [ , ]

                                             = [ , ]

                                             = [0.54 , 0.75]

Therefore, 95% confidence interval for p is [0.54 , 0.75].

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