Question

let a1 a2 · · · a10 be 10 observations with median 3. then
a. at least 5 observations are less than or equal to 3
b. at most 5 observations are less than or equal to 3
c. exactly 5 observations are less than or equal to 3
d. all observations are less than or equal to 3

Answer 1

Answer:

Option: (a) : at least 5 observations are less than or equal to 3

Step-by-step explanation:

Let $a_{(n)}$ be nth order statistics such that $a_{i+1}\geq a_{i} \:\forall \:i \in [1,9]$

Since , sample size is even .

Hence,

$Median = \frac{a_{(n/2)}+a_{(n/2+1)}}{2}\\ \\=> \frac{a_{(5)}+a_{(6)}}{2}=3 \\ \\ => a_{(5)}+a_{(6)} = 6 ,a_{(5)}\leq a_{(6)}$

Now, we get

$a_{(5)} \leq 3,a_{(6)}\geq3$

Hence , we can say at least 5 observations are less than or equal to 3.

Otherwise, if it was option B.

Contradiction : if $a_{(5)} = a_{(6)} =3$

hence , 1 to 6 are already less than or equal to 3.

If option C:

Contradiction : same as above

If option D:

Contradiction : $a_{(5)} = 2 ,a_{(6)} =4$