Question

The inside diameter of a randomly selected piston ring is a random variable with mean value 16 cm and standard deviation 0.05 cm. Suppose the distribution of the diameter is normal. (Round your answers to four decimal places.) Calculate P(15.99 ≤ X ≤ 16.01) when n = 25.

Answer 1

Answer:

P(15.99 ≤ X ≤ 16.01) = 0.1586

Step-by-step explanation:

Explanation:-

Step(i):-

Given data the inside diameter of a randomly selected piston ring is a random variable with mean value 16 cm and standard deviation 0.05 cm.

The mean of the Population 'μ' = 16cm

The standard deviation of the Population 'σ' = 0.05cm

Given 'X' be the Normal Variable

(i)  Given  X = 15.99

$Z _{1} = \frac{x-mean}{S.D} = \frac{15.99-16}{0.05} = -0.2$

(ii)   Given X = 16.01

$Z_{2} = \frac{x-mean}{S.D} = \frac{16.01-16}{0.05} = 0.2>0$

Step(ii):-

P(x₁ ≤ X ≤ x₂) = A(z₂)+A(z₁)

P(15.99 ≤ X ≤ 16.01)= P(-0.2≤ z ≤ 0.2) = A(z₂)+A(z₁)

                                                      = A(0.2)+A(-0.2)    [ A(-Z) = A(Z)]

                                                      = 0.0793 + 0.0793

                                                       = 0.1586

Conclusion:-

P(15.99 ≤ X ≤ 16.01) = 0.1586