Question

If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is

(a) Within 1.1 SDs of its mean value?


(b) Farther than 1.6 SDs from its mean value?


(c) Between 1 and 2 SDs from its mean value?

Answer 1

a.)

P(z < 1.1) = 0.8643

And

P(z < -1.1) = 0.1357

So,

Within 1.1 standard deviation from the mean

= P(-1.1 < z < 1.1)

= P(z < 1.1) - P(z < -1.1)

= 0.8643 - 0.1357

= 0.7286

b.)

P(z > 1.6) = 0.0548

And

P(z < -1.6) = 0.0548

So,

Farther than 1.6 standard deviation from the mean

= P(z > 1.6) + P(z < -1.6)

= 0.0548 + 0.0548

= 0.1096

c.)

P(z < 2) = 0.9772

And

P(z < -2) = 0.228

Therefore,

P(-2 < z < 2)

= P(z < 2) - P(z < -2)

= 0.9772 - 0.0228

= 0.9544

Similarly,

P(z < 1) = 0.8413

And

P(z < -1) = 0.1587

Therefore,

P(-1 < z < 1)

= P(z < -1) - P(z < 1)

= 0.8413 - 0.1587

= 0.6826

So,

Between 1 and 2 Standard Deviation from the mean

= P(-2 < z < 2) - P(-1 < z < 1)

= 0.9544 - 0.6826

= 0.2718