Archie is fed up with waiting in line at his local post office, and plans to contact his congresswoman. Archie knows that his complaint will not be well received without data to back it up. So to improve his case, over the next few months Archie records the waiting time for a selection of  post office visits made by him and other customers. These waiting times (in minutes) are summarized in the following frequency distribution.

Waiting time
(in minutes)	Frequency
 0to6 	
 7to13 	
 14to20 	
 21to 27	
 28to 34	
Based on the frequency distribution, using the midpoint of each data class, estimate the mean waiting time for the data Archie collected. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

To estimate the mean waiting time from a frequency distribution, we need to follow these steps:

1. **Identify the midpoints of each class**: The midpoint of each class is the average of the lower and upper boundaries of the class.
   
2. **Multiply the midpoint of each class by the frequency**: This gives us the "weighted waiting time" for each class.

3. **Sum up all the products from step 2**: This is the total of all waiting times, weighted by frequency.

4. **Divide the total weighted waiting time by the total frequency**: This gives the mean waiting time.

### Step-by-step:

#### 1. Calculate the midpoints of each class:
- For \( 0 \text{ to } 6 \), midpoint = \( \frac{0 + 6}{2} = 3 \)
- For \( 7 \text{ to } 13 \), midpoint = \( \frac{7 + 13}{2} = 10 \)
- For \( 14 \text{ to } 20 \), midpoint = \( \frac{14 + 20}{2} = 17 \)
- For \( 21 \text{ to } 27 \), midpoint = \( \frac{21 + 27}{2} = 24 \)
- For \( 28 \text{ to } 34 \), midpoint = \( \frac{28 + 34}{2} = 31 \)

#### 2. Assign frequencies for each class:
Since you haven't provided the exact frequencies, I will assign sample frequencies as placeholders:
- \( 0 \text{ to } 6 \): Frequency = \( f_1 \)
- \( 7 \text{ to } 13 \): Frequency = \( f_2 \)
- \( 14 \text{ to } 20 \): Frequency = \( f_3 \)
- \( 21 \text{ to } 27 \): Frequency = \( f_4 \)
- \( 28 \text{ to } 34 \): Frequency = \( f_5 \)

#### 3. Multiply midpoints by frequencies:
- \( 3 \times f_1 \)
- \( 10 \times f_2 \)
- \( 17 \times f_3 \)
- \( 24 \times f_4 \)
- \( 31 \times f_5 \)

#### 4. Sum up all the products and divide by the total frequency:
\[
\text{Mean waiting time} = \frac{(3 \times f_1) + (10 \times f_2) + (17 \times f_3) + (24 \times f_4) + (31 \times f_5)}{f_1 + f_2 + f_3 + f_4 + f_5}
\]

Please provide the frequencies for each class, and I'll complete the calculation for you.

Would you like more clarification on this method, or additional details?

Here are 5 related questions for further exploration:
1. What happens to the mean waiting time if one class has a much higher frequency?
2. How does this approach differ from calculating the median of a frequency distribution?
3. What are the potential limitations of using the midpoint in estimating waiting times?
4. How could Archie improve the precision of his data collection?
5. What other statistical measures could Archie use to strengthen his case?

**Tip:** Ensure that the frequencies reflect all customers' experiences to avoid sampling bias in your data collection.

Archie is fed up with waiting in line at his local post office, and plans to contact his congresswoman. Archie knows that his complaint will not be well received without data to back it up. So to improve his case, over the next few months Archie records the waiting time for a selection of post office visits made by him and other customers. These waiting times (in minutes) are summarized in the following frequency distribution.

Waiting time (in minutes) Frequency: 
0 to 6
7 to 13
14 to 20
21 to 27
28 to 34
Based on the frequency distribution, using the midpoint of each data class, estimate the mean waiting time for the data Archie collected. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

Learn how to estimate the mean waiting time using a frequency distribution based on post office waiting times. This statistical problem involves calculating midpoints for each time range and applying the weighted average formula. Suitable for students in Grades 9-11, this example helps reinforce core concepts in statistics.

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