**What Is a Confidence Interval?**

The sample statistics (or point estimates) – such as the mean, standard deviation, proportion, etc. – are used to make inference about a population based on a random sample from that population. The point estimate likely does not equal the population parameter it estimates, but should be close. The confidence interval is a range around the point estimate that has a specific probability of containing the population parameter, typically 0.95 for a 95% confidence interval. The confidence interval gives a better estimate of the population parameter of interest because it gives the idea of the range in which the population parameter is.

**Confidence Intervals for Single Means and Standard Deviations in STATISTICA**

In *STATISTICA*, you can use the *Descriptive Statistics *analysis available via the *Basic Statistics* module to find confidence intervals for a single mean or single standard deviation. To access this analysis, first open a data file, and then select the * Statistics* tab. In the

*group, click*

**Base***.*

**Basic Statistics**In the* Basic Statistics and Tables *Startup Panel, select

**and click**

*Descriptive Statistics**to display the*

**OK***dialog box. The options for the confidence intervals for the mean and standard deviation are on the*

**Descriptive Statistics***tab. You can specify the confidence level for each via the respective*

**Advanced***Interval*edit box.

You would then click the * Summary* button to get the requested statistics, which would include these confidence intervals.

**Using STATISTICA to Find a Confidence Interval for a Single Proportion**

The * Descriptive Statistics *analysis is useful for finding statistics regarding continuous data. Proportions are not continuous, but counts. Tools such as

*and*

**Frequency Tables***can find proportions. You can find a confidence interval for a single proportion using the*

**Tables and Banners***Power Analysis*module. This module is often used to calculate statistical power for a given analysis or to calculate the sample size required to attain a certain power level for a given analysis, but it can also be used to calculate, for a given analysis type, specialized confidence intervals not generally available in the general-purpose statistical packages.

**Confidence Interval for a Single Proportion Example**

In this example, researchers took a sample of 500 randomly selected subjects who completed four years of college. They found that 75 of them smoked on a regular basis. Thus, the sample proportion (often designated as *p̂*) of people who smoked and had a four-year college education is 75/500=0.15 (or 15%). If we wanted an estimate of the true proportion (usually designated as *p*) of people who smoke that have a four-year education, we could construct a confidence interval for the proportion.

The simplest and most commonly used formula for this type of confidence interval relies on approximating the binomial distribution with a normal distribution (the proportion is binomial because the person sampled either smoked or did not smoke). The formula is:

where *z₁-α⁄2* is the *1-α⁄2* percentile of the standard normal distribution; *α* is the Type I error rate and is the complement of the confidence level. Thus, for a 95% confidence level, the error *α* is 5% or 0.05.

This *z*-score can be calculated within *STATISTICA*. On the* Statistics *tab in the

*group, click*

**Base***to display the*

**Basic Statistics***Startup Panel. Select*

**Basic Statistics and Tables***.*

**Probability calculator**

Click O* K *to display the

*.*

**Probability Distribution Calculator**In the

*field, select*

**Distribution***. Select the*

**Z (Normal)***, and (*

**Inverse, Two-tailed***) check boxes. We are using*

**1-cumulative p***α*= 0.05, so enter this value for

**. Click the**

*p**button to calculate the*

**Compute****critical value (which is given in the**

*z**edit field). It is found to be 1.959964, which is commonly rounded to 1.96.*

**X**

Thus, the confidence interval for the true proportion is 0.15-1.96*sqrt[(0.15)(0.85)/500] < *p* < 0.15+1.96*sqrt[(0.15)(0.85)/500]→0.11870131 <* p* < 0.18129869.

**Finding the Confidence Interval in ****STATISTICA**

As previously mentioned, we can find this same confidence interval for a single proportion using the *Power Analysis*module in *STATISTICA*.

With any data file opened, select the * Statistics* tab. In the

*group, click*

**Advanced/Multivariate***. In the*

**Power Analysis***Startup Panel, select*

**Power Analysis and Interval Estimation***as the analysis category, and then select*

**Interval Estimation***as the analysis type.*

**One Proportion, Z, Chi-Square Test**In the* Single Proportion: Interval Estimation *dialog box, enter 0.15 for

*, 500 for*

**Observed Proportion p***, and 0.95 for*

**Sample Size (N)***.*

**Conf. Level**

Click * Compute* to calculate the confidence interval.

The* Pi (Crude)* results should match what was calculated earlier by hand as these are the estimates using the normal approximation to the binomial distribution (note that the hand calculations could be off a little due to rounding the *z*critical value to 1.96; *STATISTICA* will carry this out to more decimals for better accuracy).

The results in the* Interval Estimation *spreadsheet also include two other ways to calculate the confidence interval for a proportion – *Pi (Exact)* (the confidence intervals are the "exact, Clopper-Pearson" confidence intervals) and* Pi (Approximate)* (the confidence intervals employ a score method with a continuity correction). For more information on how these two methods are computed, see methods 4 and 5 from Robert Newcombe’s paper, Two-Sided Confidence Intervals for the Single

Proportion: Comparison of Seven Methods (1998, *Statistics in Medicine, 17*, 857-872).

**Conclusion**

Sometimes a researcher wants to estimate the true proportion of a population of interest by finding the confidence interval for that proportion. In *STATISTICA*, the *Power Analysis *module provides the means to find this estimate.