Consider a tech company that wants to offer free breakfast to its employees if their confidence interval shows it will decrease the proportion of employees who skip breakfast. Each interval shows the difference in proportion of [tex]p_1 - p_2[/tex], where [tex]p_1[/tex] represents the employees who skip breakfast before free breakfast is offered, and [tex]p_2[/tex] represents the employees who skip breakfast after free breakfast is offered. Determine if there is enough evidence to suggest that offering free breakfast results in fewer employees skipping breakfast.

Confidence intervals:

1. (0.07, 0.31)
- Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
- No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.

2. (0.04, 0.4)
- Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
- No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.

3. (−0.12, 0.2)
- Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
- No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.

To determine if providing free breakfast at a tech company results in fewer employees skipping breakfast, we need to analyze the confidence intervals provided. These intervals estimate the difference in the proportion of employees skipping breakfast before and after implementing the free breakfast.

For a confidence interval to suggest a significant reduction in the proportion of employees skipping breakfast, the entire interval should lie above zero. This indicates that the proportion of employees skipping breakfast after the intervention (free breakfast) is less than before the intervention.

Let's examine each confidence interval:

Confidence Interval: (0.07, 0.31)
- The entire interval is positive, meaning we can be confident that fewer employees are skipping breakfast when free breakfast is offered compared to when it is not. Therefore, we choose:
  "Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not."
Confidence Interval: (0.04, 0.4)
- Similar to the first interval, the entire range is positive. This also suggests fewer employees skip breakfast with the offer of free breakfast. So the choice is:
  "Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not."
Confidence Interval: (−0.12, 0.2)
- This range includes zero and a negative value, indicating the possibility of no change or even an increase in the number of employees skipping breakfast. Therefore, the choice is:
  "No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast."

The confidence intervals essentially help determine if the changes observed after providing free breakfast are due to the intervention or could happen by chance. Hence, intervals that do not include zero give us confidence that there is a significant difference. In this problem, only the first two intervals provided the evidence needed to suggest a decrease in the proportion of employees skipping breakfast.