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?

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

Step-by-step 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

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have