Question

Let X be normally distributed with mean μ = 26 and standard deviation σ = 13. [You may find it useful to reference the z table.]

a. Find P(X ≤ 0). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

b. Find P(X > 13). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

c. Find P(13 ≤ X ≤ 26). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

d. Find P(26 ≤ X ≤ 39). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

SOLUTION:

From given data,

Let X be normally distributed with mean μ = 26 and standard deviation σ = 13.

Where,

μ = 26

σ = 13.

a. Find P(X ≤ 0).

P(X ≤ 0) = P( (x-μ) / σ ≤ (0-26) / 13  )

= P( z ≤ -26 / 13  )

   = P( z ≤ -2 )

= 0.0228

b. Find P(X > 13).

P(X > 13) = 1 - [P( (x-μ) / σ ≤ (13-26) / 13  )]

=1- [ P( z ≤ -13 / 13  )]

   = 1-[P( z ≤ -1 )]

= 1- 0.1587

= 0.8413

c. Find P(13 ≤ X ≤ 26)

P(13 ≤ X ≤ 26) = P( (13-26) / 13 ≤ (x-μ) / σ ≤ (26-26) / 13  )

= P(-13 / 13 ≤ z ≤ 0 / 13  )

   = P(-1 ≤ z ≤ 0 )

= p(z ≤ 0 ) - p(z ≤ -1 )

= 0.5-0.1587

= 0.3413

d. Find P(26 ≤ X ≤ 39).

P(26 ≤ X ≤ 39) = P( (26-26) / 13 ≤ (x-μ) / σ ≤ (39-26) / 13  )

= P(0 / 13 ≤ z ≤ 13 / 13  )

   = P(0 ≤ z ≤1 )

= p(z ≤ 1 ) - p(z ≤ 0 )

= 0.8413-0.5

= 0.3413