Wednesday, May 28, 2014

Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013

This is one of the following five articles on Confidence Intervals in Excel

z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013

t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013

Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean

Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013

Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

 

Confidence Interval of

Population Proportion in 2

Steps in Excel

Confidence intervals covered in this manual will either be Confidence Intervals of a Population Mean or Confidence Intervals of a Population Proportion. A data point of a sample taken for a confidence interval of a population mean can have a range of values. A data point of a sample taken for a confidence interval of a population proportion is binary; it can take only one of two values.

Data observations in the sample taken for a confidence interval of a population proportion are required to be distributed according to the binomial distribution. Data that are binomially distributed are independent of each other, binary (can assume only one of two states), and all have the same probability of assuming the positive state.

A basic example of a confidence interval of a population proportion would be to create a 95-percent confidence interval of the overall proportion of defective units produced by one production line based upon a random sample of completed units taken from that production line. A sampled unit is either defective or it is not. The 95-percent confidence interval is range of values that has a 95-percent certainty of containing the proportion defective (the defect rate) of all of the production from that production line based on a random sample taken from the production line.

The data sample used to create a confidence interval of a population proportion must be distributed according to the binomial distribution. The confidence interval is created by using the normal distribution to approximate the binomial distribution. The normal approximation of the binomial distribution allows for the convenient application of the widely-understood z-based confidence interval to be applied to binomially-distributed data.

The binomial distribution can be approximated by the normal distribution under the following two conditions:

1) p (the probability of a positive outcome on each trial) and q (q = 1 – p) are not too close to 0 or 1.

2) np > 5 and nq > 5

The Standard Error and half the width of a confidence interval of proportion are calculated as follows:

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Margin of Error = Half Width of C.I. = z Valueα, 2-tailed * Standard Error

Margin of Error = Half Width of C.I. = NORM.S.INV(1 – α/2) * SQRT[ (p_bar * q_bar) / n]

 

Example of a Confidence Interval

of a Population Proportion

in Excel

In this example a 95 percent confidence interval of a population proportion is created around a sample proportion using the normal distribution to approximate the binomial distribution.

This example evaluates a group of shoppers who either prefer to pay by credit or by cash. A random sample of 1,000 shoppers was taken. 70% of the sampled shoppers preferred to pay with a credit card. The remaining 30% of the sampled shoppers preferred to pay with cash.

Determine the 95% Confidence Interval for the proportion of the general population that prefers to pay with a credit card. In other words, determine the endpoints of the interval that is 95 percent certain to contain the true proportion of the total shopping population that prefers to pay by credit card.

 

Summary of Problem Information

p_bar = sample proportion = 0.70

q_bar = 1 – p_bar = 1 – 0.70 = 0.30

p = population proportion = Unknown (This is what the confidence interval will contain.)

n = sample size = 1,000

α = Alpha = 1 – Level of Certainty = 1 – 0.95 = 0.05

confidence interval, statistics, excel, excel 2010, excel 2013, confidence interval of population proportion,population proportion

SE = Standard Error = SQRT[ (p_bar * q_bar) / n]

SE = SQRT[ (0.70 * 0.30) / 1000] = 0.014491

As when creating all Confidence of Proportion, we must satisfactorily answer these two questions and then proceed to the two-step method of creating the Confidence Interval of Proportion.

 

The Initial Two Questions That Must be Answered Satisfactorily

What Type of Confidence Interval Should Be Created?

Have All of the Required Assumptions For This Confidence Interval Been Met?

 

The Two-Step Method For Creating Confidence Intervals of Mean are the following:

Step 1 - Calculate the Half-Width of the Confidence Interval (Sometimes Called the Margin of Error)

Step 2 – Create the Confidence Interval By Adding to and Subtracting From the Sample Mean Half the Confidence Interval’s Width


The Initial Two Questions That Need To Be Answered Before Creating a Confidence Interval of the Mean or Proportion Are as Follows:

 

Question 1) Type of Confidence Interval?

a) Confidence Interval of Population Mean or Population Proportion?

This is a Confidence Interval of a population proportion because sampled data observations are binary: they can take only one of two possible values. A shopper sampled either prefers to pay with a credit card or prefers to pay with cash.

The data sample is distributed according to the binomial distribution because each observation has only two possible outcomes, the probability of a positive outcome is the same for all sampled data observations, and each data observation is independent from all others.

Sampled data points used to create a confidence interval of a population mean can take multiple values or values within a range. This is not the case here because sampled data observations can have only two possible outcomes: a sampled shopper either prefers to pay with credit card or with cash.

b) t-Based or z-Based Confidence Interval?

A Confidence Interval of proportion is always created using the normal distribution. The binomial distribution of binary sample data is closely approximated by the normal distribution in certain conditions.

The next step in this example will evaluate whether the correct conditions are in place that permit the approximation of the binomial distribution by the normal distribution.

It should be noted that the sample size (n) equals 1,000. At that sample size, the t distribution is nearly identical to the normal distribution. Using the t distribution to create this Confidence Interval would produce exactly the same result as the normal distribution produces.

This confidence interval will be a confidence interval of a population proportion and will be created using the normal distribution to approximate the binomial distribution of the sample data.

 

Question 2) All Required Assumptions Met?

Binomial Distribution Can Be Approximated By Normal Distribution?

The most important requirement of a Confidence Interval of a population proportion is the validity of approximating the binomial distribution (that the sampled objects follow because they are binary) with the normal distribution.

The binomial distribution can be approximated by the normal distribution sample size, n, is large enough and p is not too close to 0 or 1. This can be summed up with the following rule:

The binomial distribution can be approximated by the normal distribution if np > 5 and nq >5. In this case, the following are true:

n = 1,000

p = 0.70 (p is approximated by p_bar)

q = 0.30 (q is approximated by q_bar)

np = 700 and nq = 300

It is therefore valid to approximate the binomial distribution with the normal distribution.

The binomial distribution has the following parameters:

Mean = np

Variance = npq

Each unique normal distribution can be completely described by two parameters: its mean and its standard deviation. As long as np > 5 and nq > 5, the following substitution can be made:

Normal (mean, standard deviation) approximates Binomial (n,p)

If np is substituted for the normal distribution’s mean and npq is substituted for the normal distribution’s standard deviation as follows:

Normal (mean, standard deviation)

becomes

Normal (np, npq), which approximates Binomial (n,p)

This can be demonstrated with Excel using data from this problem.

n = 1000

n = the number of trials in one sample

p = 0.7 (p is approximated by p_bar)

p = the probability of obtaining a positive result in a single trial

q = 0.7 (q is approximated by q_bar)

q = 1 - p

np = 700

npq = 210

at arbitrary point X = 700

(X equals the number positive outcomes in n trials)

BINOM.DIST(X, n, p, FALSE) = BINOM.DIST(700, 1000, 0.7, FALSE) = 0.0275

The Excel formula to calculate the PDF (Probability Density Function) of the normal distribution at point X is the following:

NORM.DIST(X, Mean, Stan. Dev, FALSE)

The binomial distribution can now be approximated by the normal distribution in Excel by the following substitutions:

BINOM.DIST(X, n, p, FALSE) ≈ NORM.DIST(X, np, npq, FALSE)

NORM.DIST(X, np, npq, FALSE) = NORM.DIST(700,700,210,FALSE) = 0.0019

BINOM.DIST(X, n, p, FALSE) = BINOM.DIST(700, 1000, 0.7, FALSE) = 0.0275

The difference is less than 0.03 and is reasonable close. Note that this approximation only works for the PDF (Probability Density Function) and not the CDF (Cumulative Distribution Function – Replacing FALSE with TRUE in the above formulas would calculate the CDF instead of the PDF).


We now proceed to the two-step method for creating all Confidence Intervals of a population proportion. These steps are as follows:

Step 1) Calculate the Width of Half of the Confidence Interval

Step 2 – Create the Confidence Interval By Adding and Subtracting the Width of Half of the Confidence Interval from the Sample Mean


Proceeding through the four steps is done is follows:

 

Step 1) Calculate Width-Half of Confidence Interval

Half the Width of the Confidence Interval is sometimes referred to the Margin of Error. The Margin of Error will always be measured in the same type of units as the sample proportion is measured in, which is percentage. Calculating the Half Width of the Confidence Interval using the t distribution would be done as follows in Excel:

Margin of Error = Half Width of C.I. = z Valueα, 2-tailed * Standard Error

Margin of Error = Half Width of C.I. = NORM.S.INV(1 – α/2) * SQRT[ (p_bar * q_bar) / n]

Margin of Error = Half Width of C.I. = NORM.S.INV(0.975) * SQRT[ (0.7 * 0.3) / 1000]

Margin of Error = Half Width of C.I. = 1.95996 * 0.014491

Margin of Error = Half Width of C.I. = 0.0284, which equals 2.84 percent

 

Step 2 Confidence Interval = Sample Proportion ± C.I. Half-Width

Confidence Interval = Sample Proportion ± (Half Width of Confidence Interval)

Confidence Interval = p_bar ± 0.0284

Confidence Interval = 0.70 ± 0.0284

Confidence Interval = [ 0.6716, 0.7284 ], which equals 67.16 percent to 72.84 percent

We now have 95 percent certainty that the true proportion of all shoppers who prefer to pay with a credit card is between 67.16 percent and 72.84 percent.

A Excel-generated graphical representation of this confidence interval is shown as follows:

confidence interval, statistics, excel, excel 2010, excel 2013, confidence interval of population proportion,population proportion(Click On Image To See a Larger Version)

 

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