A Confidence interval for a proportion

Definition: confidence interval estimate is a range of values constructed from sample data so that
the population parameter is likely to occur within that range at a specified probability. The
specified probability is called the level of confidence. The confidence level is the overall capture
rate if the method is used many times. The sample mean will vary from sample to sample, but the
method estimate ±margin of error is used to get an interval based on each sample. C% of these
intervals capture the unknown population mean 𝜇. In other words, the actual mean will be
located within the interval C% of the time.

Confidence interval = sample mean ± margin of error

1. The population mean for a certain variable is estimated by computing a confidence

interval for that mean.

2. If several random samples were collected, the mean for that variable would be slightly
different from one sample to another. Therefore, when researchers estimate population
means, instead of providing only one value, they specify a range of values (or an interval)
within which this mean is likely to be located.

To obtain this confidence interval, add and subtract the margin of error from the sample mean.
This result is the upper limit and the lower limit of the confidence interval. The confidence
interval may be wider or narrower depending on the degree of certainty, or estimation precision,
that is required. It’s basically a range of values that likely contains the true population value
based on sample data it also gives an idea of how uncertain our estimate can be sometimes. The
confidence intervals helps account for sampling error, giving a range instead of just a single
point.
Confidence Interval Components:
1.Point Estimate: The best guess is from the sample mean or sample proportion.

2. Margin of Error: How much the estimate might vary calculated using:

● Standard error Z or
● t value

3. Confidence Level: The probability (in percentage) that the interval actually contains the true
population value.

Higher confidence = wider interval Common levels: 90%, 95%, 99%

Two Main Types of Confidence Interval:

A. Confidence Interval for a Mean

● When population standard deviation is known (use Z)

● If unknown and sample size is small, use t-distribution

B. Confidence interval for proportion

● Estimate the true proportion of sample proportion

● Gives a range where we expect the true proportion to fall with a certain level of
confidence (like 95% )

Margin of Error
1. To find a confidence interval, the margin of error must be known.

2. The margin of error depends on the degree of confidence that is required for the
estimation.

3. Typically, degrees of confidence vary between 90% and 99.9%, but it is up to the
researcher to decide.
 4. The level of confidence is represented by z*

5. It is also necessary to know the standard deviation of the variable in the population.
(Note: the population standard deviation is NOT the same as the sample standard
deviation).

6. Finally, the size of the sample n will be used to compute the margin of error.

Confidence Intervals
1. The formulas for the confidence interval and margin of error can be combined into one
formula.

Confidence Interval for Proportion:

Confidence interval for a proportion is a statistical range that estimates the true proportion of a
population based on data from a sample. It provides not just a single estimate like 60% of people
prefer a product but a range of values within which the actual population proportion is likely to
fall. This range is calculated using the sample proportion, the sample size, and a Z-score that
corresponds to the chosen confidence level (such as 90%, 95%, or 99%). The formula for a
confidence interval for proportion is:

Where P is the sample proportion, n is the sample size, and Z

is the Z-value from the standard normal distribution.
For example, if 60 out of 100 people like a product, the sample proportion is 0.60. At a 95%
confidence level, the interval might range from approximately 0.51 to 0.69, meaning we are 95%
confident that the true proportion of all people who like the product falls within that range.
Confidence intervals help us understand the reliability of an estimate and how much it might
vary due to sampling randomness. The larger the sample size, the narrower, more valuable the
confidence interval.

Confidence Interval for a Population Proportion: Assumptions

1.The binomial conditions have been met. Briefly, these conditions are:

● The sample data is the result of counts.

● There are only two possible outcomes. (We usually label one of the outcomes a “success”
and the other a “failure.”)
● The probability of success remains the same from one trial to the next.
● The trials are independent. This means the outcome on one trial does not affect the
outcome on another.

2. The values nπ and n(1- π) should both be greater than or equal to 5.

Confidence Interval for a Population Proportion- Example 1:

Team A is considering a proposal to merge with Team B. According to Team A, at least three-
fourths of the membership must approve any merger. A random sample of 2,000 current Team A
members reveals1,600 plan to vote for the merger proposal. What is the estimate of the
population proportion?

● Develop a 95 percent confidence interval for the population proportion.

● Basing your decision on this sample information, can you conclude that
● the necessary proportion of team members favor the merger.
Solution:

Confidence Interval for a Population Proportion- Example 2:

A market survey was conducted to estimate the proportion of homemakers who would recognize
the brand name of a cleanser based on the shape and the color of the container. Of the 1,400
homemakers sampled, 420 were able to identify the brand by name.

● Estimate the value of the population proportion.

● Develop a 99 percent confidence interval for the population proportion.
● Interpret your findings.

Solution:

Given Data:

● Sample size, $ n = 1400 $

● Number who recognized the brand, $ x = 420 $

● Confidence level = 99%

1. Estimate the population proportion p ˆ

The sample proportion p ˆ is given by:

 x 420
pˆ= = =0.3
n 1400

So, the estimated proportion of homemakers who recognize the brand is 0.3 (or 30%).

2. Develop a 99% confidence interval for the population proportion

Step 1: Identifying the critical value zα/2

For a 99% confidence interval, the significance level α = 1 - 0.99 = 0.01 $.

The critical value zα/2 corresponds to the z-score with 0.005 in each tail (because 0.01/2 = 0.005).

From standard normal tables,

z 0.005 ≈ 2.576

Step 2: Calculating the standard error (SE)

SE= √❑

Step 3: Calculating the margin of error (ME)

ME=z α / 2 × SE=2.576 ×0.012247 ≈ 0.03154

Step 4: Calculating the confidence interval

p ˆ ± ME=0.3 ± 0.03154

So,

Lower limit =0.3−0.03154=0.26846

Upper limit =0.3+ 0.03154=0.33154

3. Interpretation of findings
We are 99% confident that the true proportion of all homemakers who can recognize the brand
by the shape and color of the container lies between 26.8% and 33.2%.

This means if we were to repeat this survey many times, approximately 99% of the confidence
intervals constructed from those samples would contain the true population proportion.