To analyze how the birthday of the 18-year-old student affects the mean and median of the ages in the statistics class, let’s first calculate the initial mean and median of the ages provided.
Initial Ages: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18
Step 1: Calculate the Mean
The mean is calculated by summing all ages and dividing by the number of ages.
- Sum of ages: \[ 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 18 = 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 18 = 14 + 153 + 164 + 17*5 + 18 = 14 + 45 + 64 + 85 + 18 = 226 \]
- Count of ages: 13
Mean: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Ages}} = \frac{226}{13} \approx 17.38461538461538 \]
Step 2: Calculate the Median
To find the median, we need to list the ages in order (which they already are) and find the middle number.
For an odd number of ages (13), the median is the middle value. The 7th age in a sorted list will be the median.
The sorted ages are: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18
The median (7th age) is: 16.
Step 3: After Birthday Change
When the 18-year-old student has a birthday and turns 19, the new ages will be: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19.
Step 4: Recalculate the Mean
The new sum of the ages: \[ 14 + 15 + 15 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 19 = 14 + 153 + 164 + 17*5 + 19 = 14 + 45 + 64 + 85 + 19 = 227. \]
New mean: \[ \text{New Mean} = \frac{227}{13} \approx 17.46153846153846 \] Thus, the mean does increase.
Step 5: Recalculate the Median
The sorted ages after the birthday are: 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19.
The 7th number is still 16.
Thus, the median remains the same.
Conclusion:
- The mean age will increase.
- The median age will remain the same.
Therefore, the correct answer is: B. The mean age will increase, and the median age will remain the same.
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