Question

What is a confidence interval, how is it used, and why might it be important in making business decisions? Give a business example to support your answer.

Answer 1

Confidence Intervals

In statistics, a confidence interval gives the percentage probability that an estimated range of possible values in fact includes the actual value being estimated

For example, a business might estimate that a machine uses 10 lbs. of plastic for each unit of a product created. Because the machine cannot be expected to use precisely 10 lbs. per unit, a confidence interval can be created to give a range of possibilities. The company might predict that there is a 95 percent chance that the machine uses on average between 9.85 and 10.5 lbs. of plastic per unit. The confidence interval in this example is 95 percent, and the likelihood that the actual amount of plastic used is outside the estimated range is 5 percent.

importance of confidance interval:

Budget Forecasting

When a business forecasts a budget for a fiscal period, it will need to estimate both revenues and costs. If a company is significantly off the mark on either estimation, it could get in financial trouble. By using a range of possible values for revenues and costs and finding the confidence interval of those values, a business can have the information it needs to make important financial decisions.

Risk Management Because it is impossible to predict a future event with 100 percent accuracy, confidence intervals are used by businesses to manage risk. For example, if a company is 95 percent confident that sales in the next period will be between 5 million and 6 million units, there is still a 5 percent chance that they will be above or below that number.

Market Research One effect of confidence intervals in businesses is in determining the reliability of market research. Marketing is an important function for most firms, particularly when estimating their level of future sales. A company will want to have an idea of how many products it will sell in a given financial period, but cannot know the true number with certainty until after the end of the period. By collecting data from customers, past sales numbers and other sources, a company can statistically estimate the value of future sales. By using a confidence interval, the company can determine the range its sales are likely to fall.

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.

For example, a business might estimate that a machine uses 10 lbs. of plastic for each unit of a product created. Because the machine cannot be expected to use precisely 10 lbs. per unit, a confidence interval can be created to give a range of possibilities. The company might predict that there is a 95 percent chance that the machine uses on average between 9.85 and 10.5 lbs. of plastic per unit. The confidence interval in this example is 95 percent, and the likelihood that the actual amount of plastic used is outside the estimated range is 5 percent.

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:

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:

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.