Durable press cotton fabrics are treated to improve their recovery from wrinkles

Question

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 50.9 pounds and standard deviation 2.9 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 25.5 pounds and standardized deviation 1.8 pounds. A clothing manufacturer tests 5 speciments of each fabric. All 10 strength measurements are independent. (Round answers to four decimal places)

a) What 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?

Explanation / Answer

a) mean breaking strength of 5 untreated specimens =50.9

and std deviation =2.9/(5)1/2=1.297

hence P(X>50) =1-P(X<50)=1-P(Z<(50-50.9)/1.297)=1-P(Z<-0.694)=1-0.2439=0.7561

b) mean difference b/w untreated and treated =50.9-25.5=25.4

and std deviation of difference =(2.92/5+2.82/5)1/2 =1.803

hence P(X>25)=1-P(X<25)=1-P(Z<(25-25.4)/1.803)=1-P(Z<-0.2219)=1-0.4122=0.5878

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