Question

The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch.

Required:
a. What is the probability that the strength of a sample is less than 38 lb/in²?
b. If the specifications require the tensile strength to exceed 30 lb/in², what proportion of the samples is scrapped?

Answer 1

a) the probability that the strength of a sample is less than 38 lb/in² is approximately 0.9332

b) approximately 0.62% of the sample are scrapped

Explanation:

a)

What is the probability that the strength of a sample is less than 38 lb/in²?

lets X rep the paper tensile strength in pounds per square inch.

the probability that the strength of the sample is less than 38 lb/in²

i.e P( X < 38 )

so assuming that X is normally distributed with u = 35 lb/in² and α = 2 lb/in²,

so the probability expression P( X < 38 ) can be standardized as

P( X < 38 ) = P( (X-35)/2 < (38-35)/2)

= P( X < 1.5 )

from the cumulative standard normal distribution table,

we find P( X < 1.5 ) = 0.9332

so

P( X < 38 ) = P( X < 1.5 )

= 0.9332

therefore the probability that the strength of a sample is less than 38 lb/in² is approximately 0.9332

b)

If the specifications require the tensile strength to exceed 30 lb/in², what proportion of the samples is scrapped?

this can be represented as P( X < 30 )

By standardization the probability expression P( X < 30 ) can be transformed as;

P( X < 30 ) = P( (X-35)/2 < (30-35)/2)

= P( z < -2.5 )

from the cumulative standard normal distribution table,

we find P( X < -2.5 ) = 0.0062

so

P( X < 30 ) = P( X < -2.5 )

= 0.0062

therefore, approximately 0.62% of the sample are scrapped