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Confidence Limits

Confidence limit is a concept belonging to the statistics category of subject mathematics. In general, we say that confidence limits tell us about the accuracy level of the estimated mean we have calculated.

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Definition

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The upper and lower limits of the interval that is estimated over mean are known as confidence limits. This interval is known as confidence interval. These confidence intervals are at times very important to be determined because of the fact that the estimated mean always varies with each sample. With this confidence interval, we get the lower and the upper limit for the estimating of the mean. This also tells us about the level of uncertainty we have in the estimation of the true mean. If the interval is narrow, the estimate is more précised.

How to Calculate Confidence Limits

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For finding confidence limits it is important to find confidence interval. We follow the steps below for determination of confidence interval and hence the confidence limits.

1. We start with writing down of the phenomenon on which the test is to be done.

2. Then a sample is chosen form the given population to us. This is basically what will be used for gathering the data required for testing our hypothesis.

3. Sample mean and standard deviation are now calculated. We know that sample mean is the ratio of the sum of all the values given to the total number of values. Also, the standard deviation is the square root of the ratio of the sum of the squares of the difference of values and mean to the number of values.

$Mean$ = $\frac{(\sum\ x)}{n}$
$S.D.$ = $\sqrt {\frac{(\sum ((x\ –\ mean)^2))}{n}}$

4. A desired level of confidence is chosen out of the most common ones that are $90%$, $95%$ and $99%$. At times, it is also given in the problem itself.

5. The margin of error is calculated. It can be calculated using the formula

$Z_{(\frac{a}{2})}$ $\times$ $\frac{S.D.}{\sqrt{n}}$

Here, $Z_{(\frac{a}{2})}$ is the confidence coefficient with $a$ being the confidence level and $n$ as the size of the sample.

6. Now find the confidence interval using the given formula.
 
 $Mean\ \pm$   $Z_{(\frac{a}{2})}$ $\times$ $\frac{(S.D.)}{(\sqrt{n})}$

7. Once we have got the interval the confidence limits are the lower and upper bounds of the interval so obtained.

Confidence Limits for Mean

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The confidence interval for the mean is giving the information about the precision of the mean that has been determined by us. If the sample given is variable and small, then the sample mean that is determined is likely to be far from the mean population. While if the sample has some scattering and is large, then the determined sample mean is likely to be close to the mean population. For interpretation of confidence interval (limits) of mean it is important to assume that all of the values are sampled independently and randomly from a population the values of which are distributed in accordance with the Gaussian distribution.

Confidence Limits vs Confidence Interval

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A confidence interval is simply the range of values that is likely to contain the parameters of population for a sample statistic.
While confidence limits which are also called confidence bounds are just the lower bound and upper bound of the confidence interval. These values define the confidence interval while confidence interval shows a range in which the population parameters are likely to lie in.

Confidence Limits of Counts

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Suppose the data given includes a count variable. And suppose this count variable is believed to come from Poisson distribution. Also, we are computing 95% confidence interval. Then the easiest way is to approximate Poisson distribution to normal distribution with same mean and same standard deviation. With K count from a Poisson distribution, the confidence interval is
             $K\ \pm$ $Z_{(1\ –\ \frac{a}{2})}$ $\times\ \sqrt{K}$
With this we will get the confidence interval for the count and thus, the confidence limit of counts can be determined easily.
Supposedly we have several count variables. In such cases, we find the average of all the counts and then use the mean count instead of the single count in the above formula. Suppose we have $K_1$, $K_2$, $…$, $K_n$.

Then mean of counts, Mean $K$ = $\frac{(K_1\ +\ K_2\ +\ …\ +\ K_n)}{n}$

Also, confidence interval can be found via the formula
($Mean\ K$) $\pm$ $Z_{(1\ –\ \frac{a}{2})}$ $\times$ $\sqrt{\frac{(Mean\ K)}{n}}$

Example

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Example:

Suppose a person is measuring boiling temperature of a liquid and the readings noted are $102.5$, $101.7$, $100.9$, $102.2$, $103.1$ and $100.5$ on $6$ samples in degree Celsius. The sample mean evaluated is $101.82$. The standard deviation is given to be $1.2$ degrees. Find the confidence limits at $95%$ confidence level.

Solution:

The confidence interval is given by 

$Mean\ x\ \pm$ $Z_{(\frac{a}{2})}$ $\times$ $\frac{S.D.}{\sqrt{n}}$

Here mean $x$ = $101.82$, $a$ = $95%$, $S.D.$ = $1.2$, $n$ = $6$

When we substitute the values and calculate we get ($101.01$, $102.63$)
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