Question

the numbers of students in the 9 schools in a district are given below. (note that these are already ordered from least to greatest.) 204, 230, 265, 274, 329, 353, 361, 369, 387 suppose that the number 387 from this list changes to 468. answer the following. (a) what happens to the median? it decreases by \\(\square\\). it increases by \\(\square\\). it stays the same. (b) what happens to the mean? it decreases by \\(\square\\). it increases by \\(\square\\). it stays the same.

Explanation:

Part (a)

Step1: Recall median definition

For a set with \( n \) (odd) values, median is the \( \frac{n + 1}{2} \)-th term. Here, \( n = 9 \), so median is the \( \frac{9+1}{2}=5 \)-th term.

Step2: Find original median

Original data (ordered): \( 204, 230, 265, 274, 329, 353, 361, 369, 387 \). 5th term is \( 329 \).

Step3: Find new median

New data (after changing 387 to 468, still ordered): \( 204, 230, 265, 274, 329, 353, 361, 369, 468 \). 5th term is still \( 329 \).

Step1: Recall mean formula

Mean \( \bar{x}=\frac{\sum x}{n} \), where \( \sum x \) is sum of data, \( n = 9 \).

Step2: Calculate change in sum

Original value: \( 387 \), new value: \( 468 \). Change in sum: \( 468 - 387 = 81 \).

Step3: Calculate change in mean

Mean change: \( \frac{\text{Change in sum}}{n}=\frac{81}{9}=9 \). So mean increases by 9.

Answer:

It stays the same.

Part (b)