Section 9.2: Large-Sample Confidence Interval for a

Population Proportion

• Confidence Interval (CI) – for a population characteristic is an interval estimate of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the value of the characteristic is captured between the lower and upper endpoints of the interval.

• Confidence Level – associated with a confidence interval estimate is the success rate of the method used to construct the interval

Property of the Sampling Distribution of the Statistic p:

deviation standard and mean withnormalely approximat isthat

ondistributi samplinga has p statistic thelarge, is n When

n

)1(

• When n is large, approximately 95% of all samples of size n will result in a value of p that is within 1.96 standard deviation:

n

)1(96.1

n

ppp

n

ppp

n

ppp

)1(96.1

is interval for theformula dabbreviate An

)1(96.1,

)1(96.1

is for interval confidence 95%a large, is n When

The interval can be used as long as:

1. np ≥ 10 and n(1 - p) ≥ 10

2. The sample size is less than 10% of the population size if sampling is without replacement

3. The sample can be regarded as a random sample from the population of interest

Example

• Remember the example about affirmative action. We used π as a point of estimate.

• We found that π = 537/1013 = .530

• np = (1013)(.530) = 537 ≥ 10

• n(1 – p) = (1013)(.46) = 476 ≥ 10

Because both are larger than 10, we can use the large-sample interval.

)561.,499(.

031.530.

)016)(.96.1(530.1013

)470)(.530(.96.1530.

)1(96.1

n

ppp

Large-Sample Confidence Interval for πThe general formula for a confidence interval for a population proportion π when

1. p is the sample proportion from a random sample

2. The sample size n is large (np ≥ 10 and n(1 – p) ≥ 10)

3. The sample size is small relative to the population size if the sample is selected without replacement (i.e., n is at most 10% of the population size)

is

n

ppp

)1( value)critical z(

• Standard Error – of a statistic is the estimated standard deviation of the statistic

• If the sampling distribution of a statistic is (at least approximately) normal, the bound on the error of estimation B associated with a 95% confidence interval is (1.96)(standard error of the statistic).

• The sample size required to estimate a population proportion π to within an amount B with 95% confidence is

• The value of π may be estimated using prior information. In the absence of any such information, using π = .5 in this formula gives a conservatively large value for the required sample size (this value of π gives a larger n than would any other value)

296.1

)1(

B

n

Example

• A book includes a chapter on doctor-assisted suicide, caused a great deal of controversy in the medical community. Suppose that a survey of physicians is to be designed to estimate this proportion to within .05 with 95% confidence. How many primary-care physicians should be included in a random sample?

used. be should doctors 385least at of samplea Thus

16.38405.

96.125.

B

1.96)-(1n

.5 of value theuse We22