Question

A simple random sample of 2200 hospital patients admitted in a given year shows that 85.5% had an error on their medical bill. Which interval is the 95% confidence interval for the percent of all the hospital's admitted patients that year who had an error on their medical bill?

(84.0, 87.0)


(83.6, 87.4)


(84.7, 86.3)


(85.1, 85.9)

Answer 1

95% confidence interval for the percent of all the hospital's admitted patients that year who had an error on their medical bill is [84% , 87%].

Explanation:

We are given that a simple random sample of 2200 hospital patients admitted in a given year shows that 85.5% had an error on their medical bill.

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

P.Q. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample % of patients who had an error on their medical bill = 85.5%

n = sample of hospital patients = 2200

p = population percentage of all the hospital's admitted patients that year who had an error on their medical bill

Here for constructing 95% confidence interval we have used One-sample z proportion statistics.

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

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level

of significance are -1.96 & 1.96}

P(-1.96 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 1.96) = 0.95

P(
`-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.95

P(
`\hat p-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.95

95% confidence interval for p = [
`\hat p-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ,
`\hat p+1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.855-1.96 * {\sqrt{(0.855(1-0.855))/(2200) } }` ,
`0.855+1.96 * {\sqrt{(0.855(1-0.855))/(2200) } }` ]

= [0.84 , 0.87]

= [84% , 87%]

Therefore, 95% confidence interval for the percent of all the hospital's admitted patients that year who had an error on their medical bill is [84% , 87%].