The mean of 4, 5, 1, 3, 7, 4 is x. The numbers 2, 4, 3, 2, 3, y, 3 have mean x - 1 and median z. Then, y + z = ?

options

1. 4

2. 5

3. 6

4. 7

Given,

Mean of 4, 5, 1, 3, 7, 4 is x.

Sum of observations = 4 + 5 + 1 + 3 + 7 + 4 = 24

Mean (x) = $\dfrac{\text{Sum of observations}}{\text{No. of observations}} = \dfrac{24}{6}$ = 4.

Given,

Numbers 2, 4, 3, 2, 3, y, 3 have mean x - 1 = 4 - 1 = 3.

Sum of observations = 2 + 4 + 3 + 2 + 3 + y + 3 = 17 + y

Mean (x) = $\dfrac{\text{Sum of observations}}{\text{No. of observations}} = \dfrac{17 + \text{y}}{7}$

$\Rightarrow 3 = \dfrac{17 + \text{y}}{7}$

⇒ 7 × 3 = 17 + y

⇒ 21 = 17 + y

⇒ y = 21 - 17

⇒ y = 4

Observations in ascending order are = 2, 2, 3, 3, 3, 4, 4

Here, n = 7, which is odd.

By formula,

Median = $\dfrac{\text{n} + 1}{2} \text{ th observation}$

$= \dfrac{7 + 1}{2} \text{ th observation} \\[1em] = \dfrac{8}{2} \text{ th observation} \\[1em] = 4 \text{ th observation} \\[1em] = 3$

∴ z = 3.

y + z = 4 + 3 = 7.

Hence, Option 4 is the correct option.