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Applying Confidence Intervals in Business to Quantify Reliability

Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

In this lesson we will understand how to apply confidence intervals to a business application. This statistical measurement will provide a statement of confidence that sample measurements are actually representative of the entire population. The concepts are explained with the help of an example to calculate confidence intervals.

How Good is Your Sample?

In business, how can we measure the reliability of the things we create? The only foolproof way to accomplish this would be to test and analyze every single object produced, which is often impractical, if not impossible. Instead, we typically measure a small sample of objects and expect that this sample will be a representative of the entire population.

Any sample that we take for this purpose is only an approximation, and will always contain some amount of error or uncertainty. However, we can use the concept of confidence intervals to help determine how reliable we expect our sampling can be, compared to the entire population.

What Does the Confidence Interval Signify?

The confidence interval is a measure of the reliability of the sample mean compared to the actual mean. We can choose any confidence interval to express this information. If we choose 95% confidence, then the calculation says we can be 95% certain that the mean of the entire population will lie between the lower and upper range of the sample mean and standard deviation.

Another way to think about this is that if we ran through the sampling measurements 100 times, 95 of those results, expressed as the sample mean and range, are likely to contain the true mean. As you can see, this is not a guarantee of some specific range of results. Instead, confidence intervals express, in statistical terms, the confidence that our sample represents the entire population.

A Manufacturing Example

Suppose we are manufacturing baseballs, which are required to weigh between 141.75 and 148.83 grams for use in the major league. We just started using a new supplier for some of the raw materials. How confident are we that the new baseballs will fulfill our specifications?

Assume that we have randomly sampled 50 baseballs from the first batch that uses the new materials. From that sample, we have calculated an average weight of 145.59 grams, and those samples have a standard deviation of 1.67 grams. We can use the confidence interval (CI) equation:

Formula

CI = Confidence Interval %

AM = Arithmetic mean (average) of the sample data

Z-val = Z-value statistic for the associated CI

SD = Standard deviation of the sample data

N = Number of samples

We have numbers for all of these variables except the Z-val, short form of Z-value statistic. This is the actual tie between our chosen confidence interval and its associated normal distribution, expressed in standard deviations. For example, the Z-val for 95% is 1.96, since 95% of a normal distribution fits within 1.96 standard deviations of the mean. Similarly, the Z-val for 99% is 2.58. Here are the associated equations using these two confidence intervals:

Equations

If we follow through with the calculations, the 95% confidence interval range is between 145.13 and 146.05, which fits well within the specification range. With this statistical measurement, we can state with confidence that if all the baseballs we produced were weighed, we are 95% certain that the average weight would fall in that range.

The range of the 99% confidence interval is slightly larger, i.e between 144.98 and 146.20. This is well within our specifications, and we are 99% certain that the average of all the baseballs we produce would fall in that larger range. Generally, it is advisable to keep moderate number of samples (greater than 30) for confidence interval calculation to achieve good distribution. Viewed graphically, our average data would fall between 2 standard deviations at 95% confidence, or 3 standard deviations at 99% confidence.


Confidence Intervals
Confidence Interval

Sampling is an important consideration in this test. With a relatively small number of samples the use of the Z-value is not supportable, in a statistical sense. Also note the overall influence of the sample size in the computation. Everything else being equal, if we increase the sample size, we divide by a larger number which creates a smaller range. In that case the same confidence interval exists for a smaller target range. Looked at another way, we can increase our confidence interval for a stated range with additional sampling.

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