Suppose we conducted an experiment where participants are asked to solve a puzzle.  

Group 1:  The participants were required to solve the puzzle while listening to stressful countdown music and sounds.  

Group 2: The participants were required to solve the puzzle while calming piano music was played quietly in the background.  

A 95% confidence interval was calculated for the difference in the solving rates

(Proportion of Group 1 that solved the puzzle - Proportion of Group 2 that solved the puzzle),

which is given by

[-0.11, 0.23]

Which of the following interpretations is most applicable?  options:

Participants that were exposed to stressful music performed better in solving the puzzle.  The difference was statistically significant. 

Participants that were exposed to peaceful music performed better in solving the puzzle.  The difference was statistically significant. 

Although there were differences in the proportion of each group that successfully solved the puzzle, the difference was not statistically significant. 

If the confidence interval was calculated using Group 2 - Group 1, the interval would have been -0.23 to -0.11.

  

To interpret the confidence interval \([-0.11, 0.23]\) for the difference in solving rates between Group 1 (stressful music) and Group 2 (calming music), consider the following points:

### Key Facts:
1. **Confidence Interval**: A 95% confidence interval of \([-0.11, 0.23]\) implies that we are 95% confident that the true difference in solving rates between Group 1 and Group 2 lies within this range.
2. **Interpretation of the Interval**:
   - **Negative Values**: The interval includes negative values (e.g., -0.11), suggesting that it is possible that Group 2 (calming music) had a higher solving rate than Group 1 (stressful music).
   - **Positive Values**: The interval also includes positive values (e.g., 0.23), suggesting that it is also possible that Group 1 (stressful music) had a higher solving rate than Group 2 (calming music).
   - **Zero**: Importantly, the interval includes zero, meaning there is a possibility that there is no difference between the two groups.

### Conclusion:
Because the confidence interval includes zero, the difference between the groups is **not statistically significant**. This means that we do not have enough evidence to conclude that either group performed better than the other based on this experiment.

### Correct Interpretation:
The correct interpretation of the confidence interval \([-0.11, 0.23]\) is:

**"Although there were differences in the proportion of each group that successfully solved the puzzle, the difference was not statistically significant."**

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Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How would you interpret a confidence interval that does not include zero?
2. What are the implications of a confidence interval that includes negative values?
3. How would the results change if the confidence interval was narrower or wider?
4. What factors could influence the width of the confidence interval?
5. How would the interpretation change if we were interested in whether one group was *specifically* better than the other?

**Tip:** When a confidence interval includes zero, it indicates that the observed difference could be due to random chance, rather than a true effect.

Understand how to interpret confidence intervals in statistical experiments using an example of solving rates between two groups exposed to different music conditions. Learn about statistical significance and the implications of confidence interval ranges. Suitable for advanced high school and college-level students.

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