The weights of steel sheet produced by a given steel factory can be approximated

Question

The weights of steel sheet produced by a given steel factory can be approximated by a normal distribution with mean weight of 32.5 kg and a standard deviation of 2.2 kg.

1.compute the probability that the weight of a randomly selected sheet is less than 34 kg.

2. that factory requires sheet between 31.6 kg and 34.8 kg. Find the probability that a randomly selected sheet satisfies the factory's requirements.

3. compute the probability that the weight of a randomly selected sheet is more than 35 kg.

Explanation / Answer

Solution:

a) 32.5 - 34 = -1.5
-1.5 ÷ 2.2 = -0.6818
32.5 kg is -0.6818 standard deviations below the mean.

P ( Z<0.6818 )=1P ( Z<0.6818 )=10.7517=0.2483
From tables, 24.8% of the population is greater than this, so 75.2% is less.

b) 32.5 - 31.6 = 0.9 and 0.9 ÷ 2.2 = 0.4091
So 31.6 kg is 0.4091 standard deviations below the mean.
From tables, 65.9% of the population is greater than this so 34.1% is less.
34.8 - 32.5 = 2.3 and 2.3 ÷ 2.2 = 1.045
So 34.8 kg is 1.045 standard deviations above the mean.
From tables, 85.3% of the population is less.
So the percentage that's in between is 85.3 - 34.1 = 51.2%

c) 32.5 - 35 = -2.5
-2.5 ÷ 2.2 = -1.136
32.5 kg is 1.136 standard deviations below the mean.

P ( Z>1.136 )=P ( Z<1.136 )=0.8729

From tables, 87.3% of the population is greater than this, so 12.7% is less.

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