Confidence Intervals

Really??? Do you even measure your confidence. I did hear about the terms over confident and under confident in quite some conversations about people. But, do you really believe there is a certain measure as to how confident your are about a thing. I guess we do! Hurray!! Lord save us. We have a measure for everything and so much to remember. I would say otherwise. Cause, I instead of giving an exact value can now tell something in terms of confidence intervals.

Remember as a child when I was at school and had given an exam my mom used to ask me how much I would score. Me who is not as smart as my mom is(Sorry didn’t inherit a nit of her traits), always used to an exact score. Now, that idea didn’t turn out so well for me as I never scored exactly what I said. Then I put on my usual thinking cap and thought of giving a smart answer the next time. Now, my idea was to give her a range of score I might get by looking at my previous performance in that subject over a period of time. That worked somewhat miraculously for me. Guess what I always scored approximately in between the range I told her. My plan was a success. Cheers to me!

Only to realize later in life that this is what we would do when we grow up and start working on real world problems. Tell the certainty of everything with a percentage of confidence in a given range of values. The process of a giving a single value as an estimate of population parameter is known as point estimate. For example, mean value of a data is known as a point estimate. A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. It is developed by taking into consideration the point estimate and the standard error associated with it.

Confidence Interval = Point Estimate + Reliability Factor(Standard Error)

It is a popular misconception when it is told that with 95% certainty, the population mean falls between a certain range, say 100 to 200, it can take any value between the given interval. Since it is a constant value it will not change. The probability of a constant falling in between a certain range is 0 or 1. Confidence interval is the uncertainty associated with the sampling method. Suppose, we use the same sampling method for different samples and compute a different interval estimate of each sample, some interval estimates would include the population parameter others might not. A certainty of 90% means that with a confidence of 90% the interval estimates would include population parameter.

Here, we are faced with two scenarios to determine the confidence interval of a population parameter. They are:

1.) When the standard deviation of population is known:

When we ought to find a population parameter say mean of an entire population whose standard deviation is known then the confidence interval is found out by using simple random sample of size n.

2.) When the standard deviation of population is unknown:

Here the population of standard deviation is unknown which is the scenario in most of the practical cases. So, we replace the standard deviation by the estimated standard deviations which is also known as standard error. Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean is no longer normal with mean and standard deviation. Instead, the sample mean follows the t distribution with mean and standard deviation. The t distribution is also described by its degrees of freedom. For a sample of size n, the t distribution will have n-1 degrees of freedom. As the sample size n increases, the t distribution becomes closer to the normal distribution, since the standard error approaches the true standard deviation for large n.

To sum up, all the experiments we conduct to generalize a measure for all the population cannot be exactly found out. Instead the probability of it falling in a certain range is a more accurate way of giving the answer. Hope you enjoyed the read.

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Very simple terms: A confidence interval is a statistical device for saying, “I am pretty sure the true value of a number I am approximating is within this range. How sure? I am n% sure.” Where n is usually 95 or 99 and “this range” is some range of numeric values.

For example, if I ask you to estimate the number of jelly beans in a 3 gallon Jar you might reasonably say I am 90% sure the actual number is between 3,000 and 4,000 jelly beans. Without knowing it, you’ve just built a 90% confidence interval.

More accurate definition: When sampling from a population to estimate a mean a confidence interval is a range of values within which you are n% confident the true mean is included. n = some stated percentage, called a confidence level. If n = 95 then one can say, “In 95 out of 100 samples my estimated mean will fall within this stated range. Therefore, the true mean has a 95% chance of falling within this range. Conversely, there is a 5% chance that the true mean is not within this interval.”