In: Statistics and Probability
"Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is normally distributed with mean 52 pounds and standard deviation 2.6 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 28.4 pounds and standard deviation 2 pounds. A clothing manufacturer tests 3 specimens of each fabric. All 6 strength measurements are independent. (Round your answers to four decimal places.)
a) What is the probability that the mean breaking strength of the 5 untreated specimens exceeds 50 pounds?
b) What is the probability that the mean breaking strength of the 5 untreated specimens is at least 25 pounds greater than the mean strength of the 5 treated specimens?
GIVEN:
The breaking strength of untreated
fabric is normally distributed with mean pounds and standard
deviation
pounds.
The same type of fabric after
treatment has normally distributed breaking strength with mean
pounds and standard deviation
pounds.
SOLUTION:
(a) PROBABILITY THAT THE MEAN
BREAKING STRENGTH OF 5 UNTREATED SPECIMENS EXCEEDS
POUNDS:
Since the specimens are independent,
the sample mean of the breaking strengths of the 5 untreated
specimens is normally distributed with mean pounds and standard
deviation
pounds.
So the probability that the sample mean of the breaking strengths
of the 5 untreated specimens exceeds 50 pounds is
{Since
}
{From the z table, the probability value is the value corresponding to 1.7 row and 0.02 column.}
Thus the probability that
the sample mean of the breaking strengths of the 5 untreated
specimens exceeds 50 pounds is .
(b) PROBABILITY THAT THE MEAN
BREAKING STRENGTH OF 5 UNTREATED SPECIMENS IS ATLEAST POUNDS GREATER
THAN THE MEAN STRENGTH OF 5 TREATED SPECIMENS:
Since the specimens are independent,
the sample mean of the breaking strength of the 5 untreated
specimens is normally distributed with mean pounds and standard
deviation
pounds; the sample mean of the breaking strengths of the 5 treated
specimens is normally distributed with mean
pounds and standard
deviation
pounds.
Because the untreated and treated
sample means are independent as well, the difference of the
untreated minus treated sample means is normally distributed with
mean
pounds and standard deviation
pounds.
(Recall that the variance of a sum or difference of *independent* random variables is the *sum* of their variances, so it follows that the standard deviation is the square root of the sum of the squares of their standard deviations.)
So the probability that the sample mean of the breaking strengths of the 5 untreated specimens is at least 25 pounds greater than the sample mean of the breaking strengths of the 5 treated specimens is
P(difference of the untreated minus
treated sample means >= 25)
{Since
}
{From the z table, the probability value is the value corresponding to 0.9 row and 0.05 column.}
Thus the probability that
the sample mean of the breaking strengths of the 5 untreated
specimens is at least 25 pounds greater than the sample mean of the
breaking strengths of the 5 treated specimens is .