Confidence Intervals on Proportions

Intro

Since a confidence interval about a sample proportion is defined by the confidence level which determines the number of standard deviations to subtract from and add to the sample statistic, the approach is identical to that for finding a confidence interval about a mean. We find the critical values of z that give us a probability of for the belly of the distribution.

Confidence Interval Estimate for a Population Proportion p

population proportion p : ;...... X = # of successes,......N = population size.

sample proportion : ; ...... x = # of successes, ...... n = sample size.

When n is sufficiently large and if np 5 and n(1 – p) 5,
a distribution of proportions is approximately normally distributed.

The Mean and Standard Error for a Sample Proportion

When the population is finite size N and the sample size n > 5%N, we include the FPCF.

In such a case,

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Confidence Interval Limits for Proportions:

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Example: In a random sample, 136 of 400 people given a flu vaccine experienced some discomfort. Construct a 95% confidence interval for the true proportion of the population who will experience some discomfort from the vaccine.

Solution: We know

Substituting the data values in the formula:

The confidence interval limits are 0.34 – 0.046 = 0.294
and 0.34 + 0.046 = 0.386

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Finding Sample Size for a Proportion at a Given Level of Confidence

We solve the formula for sample error E = for the variable n to get:

If we don't know p, the population proportion, we can't use this formula. In such a case, we either use a sample statistic as an estimate for p or set p = 0.5 or a value near it.
If we set p = 0.5, p( 1 – p) = ¼ , which makes the formula for n:

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When finding sample size, always round up to the nearest integer.

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Example: The highway department wants to estimate the proportion of transport vehicles carrying too heavy a load. They wish to assert with 95% confidence that their error won't exceed 0.04. What sample size n do they need if:

a) they know the population proportion p for such vehicles is 0.25?
b) they have no idea what p is and have no way of finding out?

Solutions:

a) since we know p = 0.25, we substitute into the formula for n:

The sample should include 451 such vehicles.

b) since we don't know p, we use the 2nd formula for n setting p(1 – p) = ¼.

The sample should include 601 such vehicles.

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Confidence Interval Formulae

 Property Confidence Interval for Mean Confidence Interval for a Proportion c.i. limits sigma knownc.i. limits sigma unknown use t when use n – 1 degrees of freedom Erroruse z when n > 30. use t when when . Sample Size Parameters

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Practice

1. The operations manager for a large newspaper wants to determine the proportion of newspapers printed daily with a nonconforming attribute, such as excessive ink rub-off, improper page setup, missing pages, and/or duplicate pages. He selects a random sample of 200 newspapers of which 35 have some type of non-conformance. Set up a 90% confidence interval estimate about the true proportion of nonconforming papers in the population.

2. An automobile dealer wants to estimate the proportion of customers who still own the cars they purchased five years earlier. A random sample of 200 customers selected from the automobile dealer's records indicates that 82 still own cars that were purchased five years earlier. Set up a 95% confidence interval estimate of the population proportion of all customers who still own the cars five years after they were purchased.

3. An article in the Montreal Gazette suggests that firms owned by women start exporting at an early stage of development. In a survey of 254 women who had a leadership position in an exporting or export-ready business, 76 said that they began exporting at the start-up of their business and 140 took their first export steps within their two years of operation.

a) Set up a 90% confidence interval estimate of the population proportion of women who began exporting at the start-up of their business.

b) Set up a 90% confidence interval estimate of the population proportion of women who took their first export steps within two years.

4. A city assessor wants to study various characteristics of single-family houses in the city. A random sample of 70 houses reveals the following data:

· Heated area of the house (in square metres):

· 42 houses have central air conditioning

a) Set up a 99% confidence interval estimate of the population mean heated area of the houses.

b) Set up a 95% confidence interval estimate of the population proportion of houses that have central air conditioning.

5. The branch manager of an outlet (Store 1) of a large nationwide chain of pet supply stores wants to study characteristics of customers of her store. In particular, she decides to focus on two variables: the amount of money spent by customers and whether the customers own only one dog, only one cat, or more than one dog and/or one cat. The results from the sample of 70 customers are as follows:

· Amount of money spent:

· 37 customers own only a dog 26 customers own only a cat

· 7 customers own more than one dog and/or one cat
a) Set up a 95% confidence interval estimate of the population mean amount spent
in the pet supply store.
b) Set up a 90% confidence interval estimate of the population proportion
of customers who only own a cat.

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Solutions:

1. Solution:

data :
= 0.175 ± 0.044
0.131 < p < 0.219

2. Solution:

data : n = 200 x = 82
= 0.41 ± 0.07
0.34 < p < 0.48

3a. Solution:

data : n = 254 x = 76
= 0.299 ± 0.047
0.252 < p < 0.346

3b. Solution:

data : n = 254 x = 140
= 0.551 ± 0.051
0.5 < p < 0.602

4a. Solution:

152.5 < µ < 174.3

4b. Solution:

data : n = 70 x = 42
= 0.6 ± 0.11
0.49 < p < 0.71

5a. Solution:

\$ 19.18 < µ < \$ 23.50

5b. Solution:

data : n = 70 x = 26
= 0.371 ± 0.095
0.276 < p < 0.466