To analyze how the 18-year-old student turning 19 will affect the mean and median of the students' ages, we first calculate the current mean and median before the change.
Current ages: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18
Step 1: Calculate the mean:
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Add the ages together: \[ 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 18 = 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 18 = 16 \times 5 + 18 + 14 + 15 + 15 = 80 + 18 + 14 + 15 + 15 = 80 + 47 = 127 \] The total age sum is 127.
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Divide by the number of students: \[ \text{Number of students} = 13 \] \[ \text{Mean age} = \frac{127}{13} \approx 9.769230076 \]
Step 2: Identify the median:
- The median is the middle number in the sorted list. Since there are 13 students (an odd number), the median is the 7th number: Sorted ages: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18 The median is 16.
Step 3: Consider the change: an 18-year-old turns 19.
New ages list: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19
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Mean calculation after change: New sum of ages: \[ 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 19 = 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 19 = 129 \]
New mean: \[ \text{Mean age} = \frac{129}{13} = 9.923076923 \approx 9.92 \]
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Median calculation after change: Sorted ages: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19 (still the 7th number is 16) The median remains 16.
So, after analyzing the changes:
- The mean age increases because the sum of the ages increased and is calculated over the same number of students.
- The median age remains the same because the order of the middle number (16) does not change.
Therefore, the correct response is: The mean age will increase, and the median age will remain the same.