Plastic drinking straws are manufactured to fit into drinks

Lennie Davis

Lennie Davis

Answered question

2021-12-12

Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm. The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm.
(a) A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
(b) 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.

Answer & Explanation

Lynne Trussell

Lynne Trussell

Beginner2021-12-13Added 32 answers

Step 1 
(a) If the diameter of the straw is less than 3 mm, it can fit in the hole. The standard deviation is 0.25 mm, and the mean is 2.6 mm. 
The likelihood that it will fit in the hole in a beverage carton is, 
P(x<3)=P(xμσ<32.60.25) 
=P(z<0.40.25) 
=P(z<1.60) 
The probability of z less than 1.60 can be obtained using the excel formula “=NORM.S.DIST(1.60,TRUE)”. The probability value is 0.9542. 
The required probability is, 
P(x<3)=P(z<1.60) 
=0.9542 
Therefore, the probability that it fits into the hole in a drinks carton is 0.9542. 
Step 2 
(b) Let x be the number of straws fit into the holes in drinks cartons which follows binomial with sample size 500 and probability of success 0.9542. 
The mean is, 
μ=np 
=500×0.9542 
=477.1 
The standard deviation is, 
σ=np(1p) 
=500×0.9542×(10.9542) 
=4.6745 
Using a binomial model with a sample size of 500 and a success probability, let x be the number of straws that fit into the holes in beverage cartons, 
P(x480)=P(x>4800.5) [Continuity correction] 
=P(x>479.5) 
=1P(x<479.5) 
=1P(xμσ<479.5477.14.6745) 
=1P(z<0.5134) 
The probability of z less than 0.5134 can be obtained using the excel formula “=NORM.S.DIST(0.5134,TRUE)”. The probability value is 0.6962. 
The required probability is, 
P(x480)=1P(z<0.5134) 
=10.6962 
=0.3038 
Hence, the probability that at least 480 straws fit into the holes in drinks cartons is 0.3038.

Fasaniu

Fasaniu

Beginner2021-12-14Added 46 answers

Step 1
XB(500, 0.9452)N(472.6, 25.898)
579.5472.6500×0.9452
=10.9125=0.0875
500×0.9452 and 500×0.9452×(10.9452) seen one
Standardising must have sq rt.
All M marks indep cc either 479.5 or 480.5 seen
correct area i.e <0.5

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