## Confidence Intervals

(Research Starters)

A confidence interval is a statistical tool that estimates the range of values with a given probability of including the unknown, true value of a population parameter (e.g., mean, variance, correlation coefficient). Although hypothesis testing tends to be more frequently used in behavioral and social science research, in many ways, confidence intervals reveal more information about the underlying population. Confidence intervals approximate how much uncertainty is associated with the researcher's estimate of the underlying parameter. They also enable researchers to better understand how much confidence can be placed in the observed results of a quantitative research study.

According to the old adage, nothing in this life is sure except death and taxes. We see supporting evidence for this statement all around us. One can stare out the window at a dismal rain while listening to the weather report predicting sun all day. Although it

may be sunny somewhere in the area, it certainly is not outside the window. Similarly, one can read the paper predicting the victory of one political candidate at the polls only to read the next day that the opposition candidate has won. The candidate may have won in some districts but lost overall. Such phenomena can also be found in behavioral and social science research. One researcher will triumphantly find support that a theory is correct. However, when another researcher tries to replicate the study, no such support is found. In research, such phenomena can be due to a number of reasons, including the complexity of human behavior, the inadequacy of the theory, and the nature of probability and inferential statistics.

Building a theory that realistically models the real world can be a difficult task. As human beings, we are constantly flooded with data from the world around us. Some of this is irrelevant to the task at hand: I do not care at this moment that the birds are singing outside my window or that there is a plane flying over head. Other of these data are important: I need to keep track of the words that my computer transcribes in order to make sure that the voice recognition software has correctly captured what I have said and that what I have said adequately expresses what I am trying to articulate. Some of these data are only important in the background, not needed now but potentially needed later: the heat of the halogen lamp at the back of my desk is unimportant unless a flammable piece of paper (or my hand) strays too near it. In order to be able to function, we need to prioritize the data. I shut out the sounds of the outside world, concentrate on the task before me, and remain just aware enough of my surroundings that I do not accidentally hurt myself.

The same is true for data concerning human behavior. Behavior is very complex, and it can be difficult to determine which pieces of information are important when building a theory and which are not. Every time we interact with someone, either in person or through communication media, we learn another piece of information or reinforce something we already know. We tend to try to move from a position of uncertainty to one of certainty. In general, knowing "truth" is not only comforting, allowing us to feel more in control of our surroundings, but it can also help us make decisions and plan for the future. However, life does not work that way. If we are open-minded, we find that there is another piece of data that challenges our assumptions and makes us rethink our theories. Similarly, statistics do not work that way, either. Statistically significant results in a research study do not "prove" anything. Rather, statistics point us with various degrees of confidence (or lack thereof) to the conclusion that one interpretation of the results is more likely than the other. Statistics do not yield black-and-white answers; they give best guesses or scientific estimates.

Null Hypothesis

In the behavioral and social sciences, quantitative research data are most frequently analyzed using inferential statistical tools. Most of the commonly used inferential statistical tools are used to test the probability of the null hypothesis (H0) being true. A null hypothesis is the statement that there is no statistical difference between the status quo and the experimental condition. If the null hypothesis is true, then the treatment or characteristic being studied makes no difference to the end result. For example, a null hypothesis might state that peer pressure has no effect on adolescents' decisions to use drugs. The alternative hypothesis (H1), on the other hand, would state that there is a correlation between peer pressure and adolescents' drug use decisions. If the researcher accepts the null hypothesis, he or she is saying that if the data in the population are normally distributed, the results of the experiment are more than likely due to chance. By accepting the null hypothesis, the researcher concludes that peer pressure has no impact on whether or not adolescents use drugs. In order for the null hypothesis to be rejected and the alternative hypothesis to be accepted, there must be a statistical significance that the difference observed between the drug-use behavior of adolescents who experienced peer pressure to use drugs and those who did not is probably due not to chance but to the real, underlying influence of peer pressure on their decision. The statistical significance is the degree to which an observed outcome is unlikely to have occurred due to chance rather than some underlying factor.

Although the ability to accept or reject a null hypothesis gives the researcher some information about the parameters of the underlying distribution, the amount of information gained is limited.

Confidence Interval

In addition to statistical tests for hypothesis testing, there is another approach to determining the statistical significance of one's research data. A confidence interval is an estimated range of values that has a given probability of including the unknown, true value of a given population parameter, such as the mean, the variance, or the correlation coefficient. This probability is called the confidence level and is expressed as a percentage, often 95 percent, meaning that if several samples are collected from the population, the unknown, true value being sought will fall within the confidence intervals of 95 percent of the samples.

The width of the confidence interval indicates the degree of uncertainty about the parameter. The narrower the interval, the more certain the researcher can be that the estimate is valid. A wide confidence interval often means that more data are needed before conclusions can be drawn about the parameter with any degree of certainty. The taller (i.e., more leptokurtic) a distribution is, the more data points are located within the confidence interval. Likewise, the wider and flatter (i.e., more platykurtic) a distribution is, the fewer data points are located within the confidence interval (see Figure 1). When the confidence interval is larger, any deviations in the research data are less likely to be significant. A narrow confidence interval means the researcher can have a high degree of confidence in the data's statistical significance.

There are three factors used in the calculation of a confidence interval. The first of these is the obtained value of the statistic (e.g., mean, variance, correlation coefficient) of the sample. In research, one assumes that this obtained value is a good estimate of the same underlying value for the wider population, called a parameter. Confidence intervals allow the researcher to better understand how much confidence can be placed in this assumption. The second element of a confidence interval is the standard error of the measure. This can be defined as the standard deviation of the distribution of the means of all the samples. The third element is the desired confidence level. Typically, confidence intervals are calculated so that the confidence level is 95 percent, but other confidence intervals can also be calculated. A confidence interval is attached to upper and lower boundaries (values) called confidence limits.

It is important to note that a 95 percent confidence interval is not the same as saying that there is a 95 percent probability that the interval contains the population parameter. The interval either contains the parameter or it does not. The 95 percent is a statement that if a large number of samples are collected from the same population, 95 percent of these samples will contain the true parameter within...

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