If 95% confidence interval for the… - CUET UG 2025 Mathematics PYQ

CUET UG 2025 — Mathematics Statistics & Applications

2025mcqmedium

If 95% confidence interval for the population mean was reported to be 140 to 150 and $\sigma = 25$ , then size of the sample used in this study is:

[Given: $Z_{0.025} = 1.96$ ]

Official previous-year question

Held on 19 May 2025 · Verified 13 Jul 2026.

Options

A
120
B
81
C
96
D
112

Did you get this right?

Solution

The curves are $3y^2 = ax$ , $y = a$ , $a > 0$ , and the $y$ -axis.

The parabola $3y^2 = ax$ can be rewritten as:

$x = \frac{3y^2}{a}$

This parabola opens to the right and passes through the origin $(0, 0)$ .

The bounded region has:

Left boundary: $y$ -axis where $x = 0$

Right boundary: the parabola where $x = \frac{3y^2}{a}$

Bottom: $y = 0$

Top: the line $y = a$

At any height $y$ , the width of the region is:

$\frac{3y^2}{a} - 0 = \frac{3y^2}{a}$

The area is:

$\text{Area} = \int_0^a \frac{3y^2}{a} \, dy$

$= \frac{3}{a} \int_0^a y^2 \, dy$

$= \frac{3}{a} \left[\frac{y^3}{3}\right]_0^a$

$= \frac{3}{a} \left(\frac{a^3}{3} - 0\right)$

$= \frac{3}{a} \cdot \frac{a^3}{3}$

$= \frac{a^3}{a}$

$= a^2$

Therefore, the area of the region is $a^2$ square units.

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