Question

A dot plot is labeled 'Alarm Clock Set.' The horizontal axis ranges from 5 AM to 7 in increments of 30 minutes. The plot show 2 dots above 5 AM, 3 dots above 5:30, 5 dots above 6:00, 4 dots above 6:30, and 1 dot above 7.
Calculate the mean and median of the given data. Round your answers to the nearest half-hour increment as needed.

(1 point)
Responses

Mean: 5:00, Median: 6:00
Mean: 5:00, Median: 6:00

Mean: 5:00, Median: 5:00
Mean: 5:00, Median: 5:00

Mean: 6:00, Median: 5:00
Mean: 6:00, Median: 5:00

Mean: 6:00, Median: 6:00
Mean: 6:00, Median: 6:00

Answer 1

To find the mean and median of the given data, we need to first list out the times based on the dot plot data:

- 5:00 AM has 2 dots: \(5:00, 5:00\)
- 5:30 AM has 3 dots: \(5:30, 5:30, 5:30\)
- 6:00 AM has 5 dots: \(6:00, 6:00, 6:00, 6:00, 6:00\)
- 6:30 AM has 4 dots: \(6:30, 6:30, 6:30, 6:30\)
- 7:00 AM has 1 dot: \(7:00\)

Now, let's list out all the times in order:
- \(5:00, 5:00, 5:30, 5:30, 5:30, 6:00, 6:00, 6:00, 6:00, 6:00, 6:30, 6:30, 6:30, 6:30, 7:00\)

This results in 15 total time entries.

### Calculating the mean:
To calculate the mean, first convert the times to a 24-hour format for easier calculation:
- 5:00 AM = 5,
- 5:30 AM = 5.5,
- 6:00 AM = 6,
- 6:30 AM = 6.5,
- 7:00 AM = 7.

Now, we can summarize the times with their counts:
- 5:00 = 2 times
- 5:30 = 3 times
- 6:00 = 5 times
- 6:30 = 4 times
- 7:00 = 1 time

Now we multiply each time by its frequency and sum them:
\[
(5 \times 2) + (5.5 \times 3) + (6 \times 5) + (6.5 \times 4) + (7 \times 1)
\]
Calculating each term:
- \(5 \times 2 = 10\)
- \(5.5 \times 3 = 16.5\)
- \(6 \times 5 = 30\)
- \(6.5 \times 4 = 26\)
- \(7 \times 1 = 7\)

Sum:
\[
10 + 16.5 + 30 + 26 + 7 = 89.5
\]

Now, divide by the total number of entries (15):
\[
\text{Mean} = \frac{89.5}{15} \approx 5.9667
\]

Converting back to hours:
- \(5.9667\) corresponds to about \(6:00\) when rounded to the nearest half hour.

### Calculating the median:
To find the median, we list the 15 values and find the middle one:
- The ordered values are: \(5:00, 5:00, 5:30, 5:30, 5:30, 6:00, 6:00, 6:00, 6:00, 6:00, 6:30, 6:30, 6:30, 6:30, 7:00\).

The median is the 8th value in this ordered list:
- The 8th value is \(6:00\).

### Final Results:
- Mean: \(6:00\)
- Median: \(6:00\)

Thus, the answer is:
**Mean: 6:00, Median: 6:00**