Let's analyze the data and derive the necessary statistical values step by step.
### a) Modal Number of Teachers in a Car
The mode of a dataset is the number that appears most frequently. From the table, we look at the numbers of teachers per car and their corresponding numbers of cars:
- 1 teacher: 3 cars
- 2 teachers: 4 cars
- 3 teachers: 6 cars
- 4 teachers: 2 cars
Clearly, the number of cars is highest for cars carrying 3 teachers. Hence, the modal number of teachers in a car is 3.
### b) Median Number of Teachers in a Car
To find the median, we must first list all instances of the numbers of teachers in each car. This means expanding each category to show all individual occurrences:
- 1 teacher: 3 times (1, 1, 1)
- 2 teachers: 4 times (2, 2, 2, 2)
- 3 teachers: 6 times (3, 3, 3, 3, 3, 3)
- 4 teachers: 2 times (4, 4)
Combining these values, we get the sequence:
[tex]\[ 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4 \][/tex]
Next, we sort this list (though it is already sorted) and find the middle value. Since there are 15 values (an odd number), the median is the middle value, which is the 8th value in this ordered set.
The sorted list again:
[tex]\[ 1, 1, 1, 2, 2, 2, 2, \mathbf{3}, 3, 3, 3, 3, 3, 4, 4 \][/tex]
Thus, the median number of teachers in a car is 3.
### c) Mean Number of Teachers in a Car
The mean is calculated by dividing the total number of teachers by the total number of cars.
First, we calculate the total number of teachers:
[tex]\[
(1 \times 3) + (2 \times 4) + (3 \times 6) + (4 \times 2)
= 3 + 8 + 18 + 8
= 37
\][/tex]
Next, we calculate the total number of cars:
[tex]\[
3 + 4 + 6 + 2
= 15
\][/tex]
Finally, the mean number of teachers in a car is:
[tex]\[
\frac{37}{15} \approx 2.47
\][/tex]
Therefore, the mean number of teachers in a car is approximately 2.47.
To summarize:
a) The modal number of teachers in a car is 3.
b) The median number of teachers in a car is 3.
c) The mean number of teachers in a car is approximately 2.47.
