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# Confidence Interval vs Prediction Interval

Confidence Interval represents a range that the mean response is likely to fall given specified settings of the predictors

Prediction Interval represents a range that a single new observation is likely to fall given specified settings of the predictors

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Natwar Lal on 17th March 2020

Applause for all the respondents - Shashikant Adlakha, Natwar Lal

Also review the answer provided by Mr Venugopal R, Benchmark Six Sigma's in-house expert.

## Question

Q 244. Explain the difference between Confidence Interval and Prediction Interval. Also highlight when is one preferred over the other while solving business problems.

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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Confidence Interval is the interval in which the population mean is supposed to fall. Confidence Intervals are determined in all hypothesis tests as we infer something about the population from the sample.

Prediction Interval is the interval in which an individual value is supposed to fall. Prediction Intervals are determined when we use statistical tools for predictions.

Since Confidence Intervals are estimates for means, there are less chances of going wrong and hence it is smaller. On the contrary, since prediction intervals are point estimates, there are higher chances of going wrong and hence it is bigger than confidence interval.

Examples

1. Estimating the Sensex or Nifty level at the month end will be like determining Confidence Intervals, whereas, estimating the price of a particular stock at month end is like determining Prediction Intervals

2. Confidence Interval - estimating the overall sales for the product mix. Prediction Interval - estimating the sales for a specific product

Regression analysis when used for forecasting or predictions will yield both confident and prediction intervals. Usage of one over the other would depend on the output / variable that the organization is forecasting for. I would believe that most organizations work with confidence intervals, while prediction levels give them an indication of the best and the worst case scenarios.

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Difference between confidence and prediction Intervals:

Confidence Interval is used for determining an interval or range with a particular confidence or probability level, which contains the population parameter of interest such as mean, standard deviation, regression coefficients, proportions etc.with  specified settings of the predictors. A confidence  interval does not reflect the data point distribution.

For example  we design an experiment to test  HbA1c(glycosylated hemoglobin) level in diabetics. We evaluate the HbA1c level in  patients with  random blood glucose level >150 mg% and  <150 mg%. We evaluate 50 patients in each category. Statistical software  can predict the responses in these two settings. Minitab calculates a  95% confidence interval of the prediction of 7.5-8.5 % for glucose level >150 mg%. So  we can be 95% confident that this range includes the mean of HbA1c for this group of patients. But we cannot say with confidence that  95% of time, the HbA1c  values will fall in range.

Confidence intervals accounts for the sampling error, when a sample characteristics is represented to characterize the entire population. So with the increase in sample size, there is reduction in sampling error and narrowing of the confidence  interval and when sample becomes equal to the population size, confidence interval will narrow down to a single value.

Prediction interval determines the future range of a single observation with a confidence/probability, given specified settings of the predictors and enough samples. Prediction interval is a kind of confidence interval, which  used in regression analysis for prediction

Using the same settings as cited above, Minitab calculates a  95% prediction interval of 7 – 9 gm %. So  it implies that 95 % values of HbA1c with specified condition of random blood sugar > 150 mg%, will have HbA1c in this range. The prediction interval  is broader than confidence interval , because  of having sampling error plus added uncertainty in predicting individual values,  as  compared to that  of  only sampling error in confidence interval.  Prediction intervals are more commonly affected by deviation of the values from gaussian  distribution.

Use of Confidence and prediction interval in business world:

Businesses can be benefited by statistics in estimating or predicting the future events.

Material  requirement: For example, a  business would require an estimation of  raw material required,   the  amount of raw material  required can be predicted with 95% confidence. Only 5%  of time,  the material used will be out of range. We can compute both prediction and confidence intervals.

Market Research:  To estimate the sales forecast,  range of sales figure can be predicted using the data from past sales, customer data and other demographic data of niche segments. So prediction interval is appropriate in this setting.

Budget Forecasting: With  values for revenues and costs and deducing the confidence and prediction intervals of those values, businesses can take important financial decisions

Risk Management: By forecasting probable  risk events by calculating prediction interval , businesses can manage the events very well in the event of it’s occurrence.

Prediction intervals are preferred over confidence intervals, when more accurate results are desired, for example-  if it is desired to obtain a  total monthly expenditure of  organization and assume that  confidence interval falls in range of 10,000-12,000 USD. If we estimate prediction interval, it will fall in range of 9500- 12700 USD. So we can be 95% confident that total  monthly expenditure value will fall in prediction interval value ranges. So businesses are better prepared with prediction interval rather than confidence interval, which estimates monthly averages.

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Benchmark Six Sigma Expert View by Venugopal R

Confidence interval is an estimated interval calculated from a set of observed data within which the a population parameter (e.g. Mean) is expected to fall with a given confidence level (e.g. 95%). It is to be remembered that the confidence interval is used for estimating the position of the population mean and not an individual value from the population.

Predictive interval is an estimated interval within which an individual future value from a population is ‘predicted’ to fall with a certain probability.

Confidence intervals give rise to a degree of certainty / uncertainty with respect to a sampling method. It provides the limits within which a population parameter will be contained. The mean value of a sample taken from a population provides a ‘point’ estimate. Imagine a large lot of apples and we need to estimate the mean weight of the apples in the population. If we take a random sample from the population and measure the mean value as 300 grams, this value is a point estimate of the population and will be subjected to uncertainties. While we may not be able to obtain the real value of the population mean unless we weigh all the apples, we can make the point estimate practically more useful by providing confidence intervals around it within which the mean population is expected to fall with a specified confidence level. This is possible by using the sample mean, sample standard deviation and assuming normal distribution.

One of the common applications of confidence intervals is during the tests for significance for means.

A common misconception about the Confidence Interval is that it is sometimes wrongly interpreted that they represent the percentage of individual values that fall within them 95% of the time (if the confidence level is at 95%).

Coming to Prediction intervals, as defined earlier, they represent the intervals for individual future value from the population. Since the variation of the individual values will be much larger than the mean values, the Prediction Intervals will be wider than the confidence intervals. Prediction intervals are usually used during regression analysis. Prediction intervals are preferred for many situations than the confidence intervals, since they provide the estimates for an individual observation rather than unobservable population parameter.

The below graph shows a fitted regression plot depicting the Confidence Intervals (green inner dotted lines) and the Prediction Intervals (violet outer dotted lines)

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Both Shashikant and Natwarlal have provided good answers. Natwarlal has differentiated the use of CI and PI clearly with his examples. For that reason, Natwarlal is the winner for this question.

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