# 8.3.1 - Confidence Intervals

8.3.1 - Confidence IntervalsRecall the general form of a confidence interval...

\(sample\ statistic\pm\underbrace{(multiplier)\ (standard\ error)}_{\textbf{margin of error}}\).

The formula for constructing a confidence interval for the difference in paired means is almost identical to the formula for constructing a confidence interval for one mean. Note that the only change is the subscript d which stands for difference.

Confidence Interval for the Difference Between Two Paired Means

\(\underbrace{\overline{x}_d}_{\text{sample statistic}} \pm \overbrace{t^*}^{\text{multiplier}} \underbrace{\left(\dfrac{s_d}{\sqrt{n}}\right)}_{\text{standard error}}\)

\(t^*\) is the multiplier with \(df = n-1\)

# 8.3.1.1. - Example: Change in Knowledge

8.3.1.1. - Example: Change in KnowledgeAn educational research study is designed so that participants complete a measure of demonstrated knowledge twice. The researcher wants to estimate the change in scores from the first to second administrations (i.e., pre- and post-test). Data are paired by participant. The researcher subtracted pre-test scores from the post test scores and found a mean increase of 6.560 with a standard deviation of 3.867 for \(n=100\). She wants to construct a 95% confidence interval for the mean difference.

First, we'll find the appropriate multiplier.

\(df=n-1=100-1=99\)

For a 95% confidence interval: \(t_{df=99}=1.984\)

\(6.560 \pm 1.984 \left(\frac{3.867}{\sqrt{100}}\right)=6.560 \pm 0.767=[5.793, 7.327]\)

We are 95% confident that the difference between post- and pre- test scores is between 5.793 and 7.327.

Data from Zimmerman, W. A. (2015). Impact of Instructional Materials Eliciting Low and High Cognitive Load on Self-Efficacy and Demonstrated Knowledge (Unpublished doctoral dissertation). The Pennsylvania State University, University Park, PA.