Peter started running 10-kilometer races two years ago. He recorded his race times for the first twelve races he completed. Study the race times given (in hours and minutes) below and answer the questions that follow.

\[
\begin{array}{cccccc}
1:12 & 1:08 & 1:11 & 1:04 & 1:02 & 1:07 \\
0:59 & 1:06 & 1:02 & 0:59 & 1:02 & 0:57 \\
\end{array}
\]

Use the information above to answer the questions that follow:

7.1.1 Determine the median time of the given data. (3 points)

7.1.2 Peter's mean race time after thirteen races was 1:03. Calculate Peter's race time for his thirteenth race.

7.1.3 Write down the probability that Peter will finish his next race (of the same distance) in less than 40 minutes. (2 points)

7.1.4 According to his last race results, Peter's race time falls in the [tex]$60^{\text{th}}$[/tex] percentile of male runners of his age. What does this mean? (2 points)

7.1.5 The given box and whisker diagram shows a summary of Peter's race time data, in minutes, over a period of two years. Compare Peter's results in the first 12 races to the results summarized in the box and whisker diagram and give reasons why it can be said that he improved his times. (4 points)

[15 points]

Sure, let's tackle each part of the problem step by step.

### 7.1.1: Determine the median time of the given data.

First, let's convert all the given race times to minutes for easier calculation:
- 1:12 = 72 minutes
- 1:08 = 68 minutes
- 1:11 = 71 minutes
- 1:04 = 64 minutes
- 1:02.8 = approx. 63 minutes (rounding for simplicity)
- 1:07 = 67 minutes
- 0:59 = 59 minutes
- 1:06 = 66 minutes
- 1:02 = 62 minutes
- 0:59 = 59 minutes
- 1:02 = 62 minutes
- 0:57 = 57 minutes

Now, let's sort these times:
57, 59, 59, 62, 62, 63, 64, 66, 67, 68, 71, 72

For a list with 12 numbers, the median is the average of the 6th and 7th numbers in the sorted list:
- 6th number: 63
- 7th number: 64

Median = (63 + 64) / 2 = 63.5 minutes

### 7.1.2: Calculate Peter's race time for his thirteenth race given the mean.

If the mean race time after 13 races is 63 minutes, let's find out the thirteenth race's time.

1. Calculate the total time for the first 12 races:
Sum = 72 + 68 + 71 + 64 + 63 + 67 + 59 + 66 + 62 + 59 + 62 + 57 = 750 minutes

2. The total time for 13 races should be 13 * 63 = 819 minutes (since mean = 63 minutes).

3. The time for the thirteenth race is:
Thirteenth race time = 819 - 750 = 69 minutes

### 7.1.3: Probability that Peter will finish his next race in less than 40 minutes.

Based on the 12 given times, none of Peter's race times were under 40 minutes. Therefore, the probability of finishing under 40 minutes, based on this data, would be 0 out of 12, which is:

Probability = 0

### 7.1.4: Explanation of the 60th percentile.

If Peter's last race result is in the 60th percentile, this means his race time was quicker than 60% of male runners of his age group. In other words, he performed better than the majority and is above average in his group for that race.

### 7.1.5: Compare Peter's results in the first 12 races to the box-and-whisker diagram.

Without the specific diagram, we can generally say that if the box-and-whisker diagram shows a lower median time or reduced interquartile range compared to his previous races, this indicates improvement. It would mean that Peter's times are becoming faster on average and more consistent (less variability).