Question

A soft drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least 90 pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of 97 psi and a standard deviation of 3.4 psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the production process to verify the claim. The bottler randomly selects 42 bottles from the last large batch produced, measures the internal pressure of each and finds the mean pressure for the sample to be 1.5 psi below the process mean cited by the vendor.

a) assuming that the vendor is correct in his claim, what is the probability of obtaining a sample mean this far or farther below the process mean?

b) If the standard deviation were 3.4 psi as claimed, but the mean was 95 psi, what is probability of obtaining a sample mean of 95.5 or below?

c) if the process mean were 97 psi as claimed, but the standard deviation was 4.8 psi, what is the probability of obtaining a sample mean of 95.5 psi or below?

Answer 1

(a)

n = Sample Size = 42

SE = /

= 3.4/ = 0.5246

Z = - 1.5/0.5246 = - 2.8593

table of Area Under Standard Normal Curve gives area = 0.4979

So,

Probability of obtaining a sample mean this far = 0.5 -0.4979 = 0.0021

So,

Answer is:

0.0021

(b)

Z = (95.5 - 95)/0.5246 = 0.9531

Table gives area = 0.3289

So

P(<95.5) = 0.5 + 0.3289 = 0.8289

So,

Answer is:

0.8289

(c)
SE = 4.8/ = 0.7407

Z = (95.5 - 97)/0.7407 = - 2.0251

Table gives area = 0.4788

So,

P(<95.5) = 0.5 - 0.4788 = 0.0212

So,

Answer is:

0.0212