Confidence intervals are often misinterpreted. The logic behind them may be a bit confusing. Remember that when we're constructing a confidence interval we are estimating a population parameter when we only have data from a sample. We don't know if our sample statistic is less than, greater than, or approximately equal to the population parameter. And, we don't know for sure if our confidence interval contains the population parameter or not.
The correct interpretation of a 95% confidence interval is that "we are 95% confident that the population parameter is between X and X."
Example: Correlation Between Height and Weight Section
At the beginning of the Spring 2017 semester a sample of World Campus students were surveyed and asked for their height and weight. In the sample, Pearson's r = 0.487. A 95% confidence interval was computed of [0.410, 0.559].
The correct interpretation of this confidence interval is that we are 95% confident that the correlation between height and weight in the population of all World Campus students is between 0.410 and 0.559.
Example: Seatbelt Usage Section
A sample of 12th grade females was surveyed about their seatbelt usage. A 95% confidence interval for the proportion of all 12th grade females who always wear their seatbelt was computed to be [0.612, 0.668].
The correct interpretation of this confidence interval is that we are 95% confident that the proportion of all 12th grade females who always wear their seatbelt in the population is between 0.612 and 0.668.
Example: IQ Scores Section
A random sample of 50 students at one school was obtained and each selected student was given an IQ test. These data were used to construct a 95% confidence interval of [96.656, 106.422].
The correct interpretation of this confidence interval is that we are 95% confident that the mean IQ score in the population of all students at this school is between 96.656 and 106.422.