Ten people in a room have an average height of 167.64cm. An 11th person who is 195.58cm tall enters the room. Which one of the following statements is correct?

Question

BlackTom AI · Accepted Answer

We start by noting the given data and what changes when the 11th person enters. 
- Ten people have an average height of 167.64 cm, so their total height is 10 × 167.64 = 1676.4 cm. 
- The 11th person is 195.58 cm tall, which increases the overall total to 1676.4 + 195.58 = 1871.98 cm, giving a new mean of 1871.98 / 11 ≈ 170.13 cm. 
- The median depends on the order statistics of the data. With 10 existing measurements, the median is the average of the 5th and 6th smallest values. After adding an 11th value, the median becomes the 6th smallest value in the new set of 11 values (not the average of two values).

Option A: “The median height will definitely decrease”
- This claim is unlikely because the new value (195.58 cm) is higher than the existing heights. Adding a tall value to a sorted list tends to push the median upward or, in some cases, leave it the same, but it cannot cause a definite decrease. Therefore this option is not correct.

Option B: “The median height may stay the same”
- This is plausible. If the 6th smallest value in the original 10 remains the 6th smallest after adding the tall person (for instance, if the new value lands above the current top half), the 6th position in the 11-number set could be the same as the previous central value (or align with the previous 6th value in certain distributions). In other words, the median does not have to change, and there are realistic data configurations where it remains the same. This makes the statement feasible.

Option C: “The median height will definitely exceed the mean height”
- The mean after adding the tall person is about 170.13 cm. The original 10-number distribution’s 6th value (the new median could be this value) might be below, around, or above 170 cm depending on the actual data. Since the mean is a fixed numeric value here and the median is not guaranteed to exceed it, this option asserts an certainty that isn’t warranted by the information given. Therefore this option is not correct.

Option D: “The median height will definitely increase”
- While adding a tall value can push the median upward in some scenarios, there are realistic possibilities where the 6th smallest value after insertion remains the same as before (as discussed in option B). Hence it is not guaranteed that the median will increase. This option is too strong.

Summary of reasoning across options: the only statement that can be true without requiring specific, exact knowledge of the 5th and 6th values is that the median may stay the same, given the tall entrant could land above the current middle positions and not alter the 6th order statistic. The other options claim definite changes (decrease or increase) or definite exceeding of the mean, which cannot be asserted from the information provided.

MATH1062/1005/1023 (ND) MATH1062/1005 Data Quiz 3

Ten people in a room have an average height of 167.64cm. An 11th person who is 195.58cm tall enters the room. Which one of the following statements is correct?

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