To determine the intervals during which the average time spent on hold increased, we need to analyze the changes in the average hold time from week to week.
Let's look at the average hold times given for each week:
- Week 1: 7 minutes
- Week 2: 5 minutes
- Week 3: 3 minutes
- Week 4: 2 minutes
- Week 5: 3 minutes
- Week 6: 5 minutes
First, we calculate the differences in hold times between consecutive weeks:
- Week 2 - Week 1: [tex]\(5 - 7 = -2\)[/tex] minutes (a decrease)
- Week 3 - Week 2: [tex]\(3 - 5 = -2\)[/tex] minutes (a decrease)
- Week 4 - Week 3: [tex]\(2 - 3 = -1\)[/tex] minutes (a decrease)
- Week 5 - Week 4: [tex]\(3 - 2 = 1\)[/tex] minute (an increase)
- Week 6 - Week 5: [tex]\(5 - 3 = 2\)[/tex] minutes (an increase)
Now, we need to look at the specified intervals and determine if the average hold time increased in those intervals:
1. Week 1 to Week 3:
- Calculate the total change from Week 1 to Week 3:
[tex]\[
(Week 2 - Week 1) + (Week 3 - Week 2) = -2 + (-2) = -4 \text{ minutes (a decrease)}
\][/tex]
2. Week 1 to Week 4:
- Calculate the total change from Week 1 to Week 4:
[tex]\[
(Week 2 - Week 1) + (Week 3 - Week 2) + (Week 4 - Week 3) = -2 + (-2) + (-1) = -5 \text{ minutes (a decrease)}
\][/tex]
3. Week 4 to Week 6:
- Calculate the total change from Week 4 to Week 6:
[tex]\[
(Week 5 - Week 4) + (Week 6 - Week 5) = 1 + 2 = 3 \text{ minutes (an increase)}
\][/tex]
4. Week 3 to Week 5:
- Calculate the total change from Week 3 to Week 5:
[tex]\[
(Week 4 - Week 3) + (Week 5 - Week 4) = -1 + 1 = 0 \text{ minutes (no change)}
\][/tex]
From this analysis, we can conclude that the only interval during which the average time spent on hold increased is:
week 4 to week 6.
