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What are the 3 common confidence level?

The most common confidence levels are 90%, 95% and 99%. The following table contains a summary of the values of corresponding to these common confidence levels. (Note that the"confidence coefficient" is merely the confidence level reported as a proportion rather than as a percentage.)

online.stat.psu.edu - S.2 Confidence Intervals | STAT ONLINE

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The previous example illustrates the general form of most confidence intervals, namely:

$ ext{Sample estimate} pm ext{margin of error}$

The lower limit is obtained by:

$ ext{the lower limit L of the interval} = ext{estimate} - ext{margin of error}$

The upper limit is obtained by:

$ ext{the upper limit U of the interval} = ext{estimate} + ext{margin of error}$ Once we've obtained the interval, we can claim that we are really confident that the value of the population parameter is somewhere between the value of L and the value of U. So far, we've been very general in our discussion of the calculation and interpretation of confidence intervals. To be more specific about their use, let's consider a specific interval, namely the "t-interval for a population mean µ."

(1-α)100% t-interval for the population mean (mu)

If we are interested in estimating a population mean (mu), it is very likely that we would use the t-interval for a population mean (mu). t-Interval for a Population Mean The formula for the confidence interval in words is: $ ext{Sample mean} pm ( ext{t-multiplier} imes ext{standard error})$ and you might recall that the formula for the confidence interval in notation is: $ar{x}pm t_{alpha/2, n-1}left(dfrac{s}{sqrt{n}} ight)$

Note that:

the " t-multiplier ," which we denote as (t_{alpha/2, n-1}), depends on the sample size through n - 1 (called the " degrees of freedom ") and the confidence level ((1-alpha) imes100%) through (frac{alpha}{2}). ," which we denote as (t_{alpha/2, n-1}), depends on the sample size through n - 1 (called the " ") and the confidence level ((1-alpha) imes100%) through (frac{alpha}{2}). the " standard error ," which is (frac{s}{sqrt{n}}), quantifies how much the sample means (ar{x}) vary from sample to sample. That is, the standard error is just another name for the estimated standard deviation of all the possible sample means. ," which is (frac{s}{sqrt{n}}), quantifies how much the sample means (ar{x}) vary from sample to sample. That is, the standard error is just another name for the estimated standard deviation of all the possible sample means. the quantity to the right of the ± sign, i.e., " t -multiplier × standard error ," is just a more specific form of the margin of error. That is, the margin of error in estimating a population mean µ is calculated by multiplying the t-multiplier by the standard error of the sample mean. ," is just a more specific form of the margin of error. That is, the margin of error in estimating a population mean µ is calculated by multiplying the t-multiplier by the standard error of the sample mean. the formula is only appropriate if a certain assumption is met, namely that the data are normally distributed.

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Clearly, the sample mean (ar{x}) , the sample standard deviation s, and the sample size n are all readily obtained from the sample data. Now, we just need to review how to obtain the value of the t-multiplier, and we'll be all set.

How is the t-multiplier determined?

As the following graph illustrates, we put the confidence level $1-alpha$ in the center of the t-distribution. Then, since the entire probability represented by the curve must equal 1, a probability of α must be shared equally among the two "tails" of the distribution. That is, the probability of the left tail is $frac{alpha}{2}$ and the probability of the right tail is $frac{alpha}{2}$. If we add up the probabilities of the various parts $(frac{alpha}{2} + 1-alpha + frac{alpha}{2})$, we get 1. The t-multiplier, denoted (t_{alpha/2}), is the t-value such that the probability "to the right of it" is $frac{alpha}{2}$: It should be no surprise that we want to be as confident as possible when we estimate a population parameter. This is why confidence levels are typically very high. The most common confidence levels are 90%, 95% and 99%. The following table contains a summary of the values of (frac{alpha}{2}) corresponding to these common confidence levels. (Note that the"confidence coefficient" is merely the confidence level reported as a proportion rather than as a percentage.)

online.stat.psu.edu - S.2 Confidence Intervals | STAT ONLINE

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