Question

A normal distribution has mean 52 and variance 64. Obtain estimated limits which include exactly middle 25 % of the observations.

Answer 1

Here, µ = 52; σ² = 64

∴ σ = 8 .

Suppose, the limits that include exactly middle 25% of the observations is x₁ to x₂.

P[x₁ ≤ X ≤ µ] = 0.125
∴ P[z₁ ≤ Z ≤ 0] = 0.125

From the table, for area 0.1217, z₁ = – 0.31 and for area 0.1255,

z₁ = – 0.32 and for area \(\frac{0.1217+0.1255}{2}\) = 0.1236,

z₁ = \(\frac{(−0.31)+(−0.32)}{2}\) = – 0.315

(∵ z₁ is on left side.)

According to above calculation, 0.125 is near to 0.1255, so we will take z₁ = -0.32.

Now, z₁ = \(\frac{x_1−52}{8}\)

∴ – 0.32 = \(\frac{x_1−52}{8}\)

∴ – 0.32 × 8 = x₁ – 52

∴ 52 – 2.56 = x₁

∴ x₁ = 49.44

P[µ ≤ X ≤ x₂] = 0.125
∴ P[0 ≤ Z ≤ z₂] = 0.125 (∵ z₂ = 0.32 ( vz2 is on right side.)

Now, z₂ = \(\frac{x_2−μ}{σ}\)

∴ 0.32 = \(\frac{x_2−52}{8}\)

∴ 0.32 × 8 = x₂ – 52

∴ 52 + 2.56 = x₂

∴ x₂ = 54.56

Hence, the limits which include exactly middle 25% of the observations of a normal distribution obtained is 49.44 to 54.56.