The Confidence Interval

In research, it is sometimes impossible to study an entire population and be able to make conclusions due to costs and other factors involved. So one chooses to take a random sample from the population and computes a statistic from it with a purpose of approximating a population parameter. The statistic in this case is the point estimate of the population parameter. A point estimate can be defined as a single value that is calculated from sample data in order for it to serve as a best estimate for an unknown population parameter. A point estimate by itself has restricted helpfulness since it does not really tell us the uncertainty associated with the estimate. It does not reveal how far our estimated statistic is from the population parameter. For instance if we are to estimate our population mean from the chosen sample, then we should ask ourselves how confident are we that the population mean lies within the sample statistic? To answer the question and be certain then the confidence interval should be computed. It provides more information than the point estimates (BROWN, 2013).

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The confidence interval can be defined as estimated range of values that are calculated from a given set of sample data and this range of values is likely to include the unknown population parameter being estimated. Researchers use the confidence interval to explain the amount of uncertainty associated with a given sample statistic of a population parameter. A definite percentage (confidence level) of the intervals will have the unknown population parameter if several independent samples are picked from similar population and then the confidence interval of each sample is computed (Liu & Chacko, 1992). The range and width of the computed confidence interval gives us the visual impression of how certain or precise we are about the unknown parameter. If we have a wide range of the interval then it means we need to collect more data before anything definite can be concluded on the unknown parameter. The confidence intervals are always constructed from a certain confidence level selected by the user for instance 95%, 90%, 99% among others.

Confidence level is defined as the probability value () that is related to a confidence interval. This implies that for any given time if similar population is sampled on several occasions and on each occasion interval estimates are constructed then the resulting intervals might bracket the true population parameter in roughly 95% of the cases. If we state a confidence interval at level then we can just work it out as the inverse of significance level. For example if say=0.05= 5% then the confidence can be illustrated as equal to =0.95 or confidence level of 95%. Confidence levels can be interpreted as for instance 90% confidence level means that 90% of the intervals include the true population parameter. Confidence intervals are also associated with limits which are upper and lower boundaries of the interval. The limits define the range of the confidence interval (Simundic, 2008).

As statisticians strive to estimate population parameters from sample data, two methods are involved which include the point estimation (involves a single value) and interval estimation (involves two values within which a population parameter lies). The best point estimate for the population mean () is the sample mean (). It is always not a question of best but rather unbiased estimate. And an estimate is always unbiased if the population parameter is equal to its expected value. Therefore the sample mean () is the best point estimate of the population mean () because E () = (Rodriguez, 2013).

To compute the confidence interval then the knowledge about what its components are should be attained. The confidence interval is made up of three parts which include the confidence level, the statistic and the margin of error. The margin of error and the statistic define the interval estimate that explains the preciseness of the method. The range of values below or above the sample statistic in a confidence interval is known as the margin of error. Therefore the interval estimates of a confidence interval are computed as sample statisticmargin of error. The confidence intervals are usually preferred compared to point estimates because of their indication of the uncertainty of the estimate and also their precision (Riffenburgh, 2006).

In the research where I am investigating whether the age difference plays a critical role in the method used to treat the infectious disease in patients at NCLEX memorial Hospital, the point estimate of the population mean age can be computed as follows;

The unbiased point estimate for the population mean of the ages of patients is therefore approximately 62 years (Liu & Chacko, 1992).

To construct a 95% confidence interval for the population mean with unknown mean and variance then the following steps were evaluated;

First, identification of the sample statistic which in my case is the sample mean () and is equal to 61.817.

Secondly, selection of the confidence level which in our case 95% is used.

Finally, calculation of the margin of error; which is given by critical value*standard error of statistic. The critical value is read from the Z-tables since the data is normally distributed. The z-value is 1.96.

Therefore the 95% confidence interval for the population mean age is;

Therefore we are 95% certain that the population mean of the ages of patients lie between the lower confidence limit (59.56) and the upper confidence limit (64.08).

The point estimate is just calculated the same way as above;

The unbiased point estimate for the population mean of the ages of patients is therefore approximately 62 years.

To construct a 99% confidence interval for the population mean with unknown mean and variance, I followed the steps below;

First, the sample statistic was identified which in my case it is the sample mean () and is equal to 61.817.

Second, confidence level was selected which in this case 99% confidence level is used.

Finally, I calculated the margin of error which is given by critical value*standard error of statistic. The critical value is read from the Z-tables since the data is normally distributed. The z-value is 2.58.

Therefore the 99% confidence interval for the ages is;

Therefore we are 99% certain that the population mean of the ages of patients lie between the lower confidence limit (58.84) and the upper confidence limit (64.79).

Comparing the 99% and the 95% confidence intervals, it can be easily noticed that the 99% confidence level widens the interval more compared to the 95% confidence level. This is because we are trying to be more confident that the confidence interval contains the population mean of the ages of patients (Rodriguez, 2013).

Therefore we can easily conclude that for one to be more confident that the estimated populations mean lies in a certain interval then we need to be less precise by increasing the confidence level as seen in the example above.

References

BROWN, S. (2013). SOME GENERALISATIONS OF CONFIDENCE INTERVAL ESTIMATES OBTAINED FROM INTERVAL ARITHMETIC. Bjbabe, IV(7), 59-63. http://dx.doi.org/10.7904/2068-4738-iv(7)-59

Liu, S., & Chacko, G. (1992). Decision-Making under Uncertainty: An Applied Statistics Approach. Technometrics, 34(2), 243. http://dx.doi.org/10.2307/1269267

Riffenburgh, R. (2006). Statistics in medicine. Amsterdam: Elsevier Academic Press.

RodrAguez, J. (2013). Mean estimation through proportion estimation. Journal Of Nonparametric Statistics, 26(2), 207-217. http://dx.doi.org/10.1080/10485252.2013.852193

Simundic, A. (2008). Confidence interval. Biochemia Medica, 154-161. http://dx.doi.org/10.11613/bm.2008.015