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# Chapter 5, Problem 40

**Chapter 5, Problem 40**

A supplier contract calls for a key dimension of a part to be between 1.96 and 2.04 centimeters. The supplier has determined that the standard deviation of the process, which is normally distributed, is 0.10 centimeter.

a. If the actual mean of the process is 1.98, what fraction of parts meet specifications?

b. If the mean is adjusted to 2.00. What fraction of parts meet specifications?

c. How small must the standard deviation be to ensure that no more than 2% of parts are nonconforming, assuming the mean is 2.00?

Specifications: Min length=1.96cm

Max length=2.04cm

Actual old mean= 1.98cm

St Dv = 0. cm

New Mena= 2.00 cm

**a. If the actual mean of the process is 1.98, what fraction of parts meet specifications?**

Z (1.96) = (1.96 – 1.98)/0.1 = -0.200

Z (2.04) = (2.04 – 1.98)/0.1 = 0.600

P (1.96 < X < 1.98) = P (-0.2 < Z < 0.6)

= Normal cdf (-0.2, 0.6)

= 0.3974

**b. If the mean is adjusted to 2.00. What fraction of parts meet specifications?**

b) Z (1.96) = (1.96 – 2.00)/0.1 = -0.400

Z (2.04) = (2.04 – 2.00)/0.1 = 0.400

P (1.96 < X <2.00) = P (-0.4 < Z < 0.4)

= Normal cdf (-0.4, 0.4)

= 0.3300

c. **How small must the standard deviation be to ensure that no more than 2% of parts are nonconforming, assuming the mean is 2.00?**

Fraction of parts that will meet the specifications must be at least 98%

P(1.96 < X < 2.04) 0.9800

New StDev will be 0.0172 cm