Math Problem Statement

An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inch. The lower and upper specification limits under which the ball bearings can operate are 0.73 inch and 0.77 ​inch, respectively. Past experience has indicated that the actual diameter of the ball bearings is approximately normally​ distributed, with a mean of 0.754 inch and a standard deviation of 0.007 inch. Complete parts​ (a) through​ (e) below. Question content area bottom Part 1 a. What is the probability that a ball bearing is between the target and the actual​ mean?    enter your response here ​(Round to four decimal places as​ needed.) Part 2 b. What is the probability that a ball bearing is between the lower specification limit and the​ target?    enter your response here ​(Round to four decimal places as​ needed.) Part 3 c. What is the probability that a ball bearing is above the upper specification​ limit?    enter your response here ​(Round to four decimal places as​ needed.) Part 4 d. What is the probability that a ball bearing is below the lower specification​ limit?    0.0003 ​(Round to four decimal places as​ needed.) Part 5 e. Of all the ball​ bearings, 91​% of the diameters are greater than what​ value?    0.745 inch ​(Round to three decimal places as​ needed.)

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Normal Distribution
Z-scores

Formulas

Z = (X - μ) / σ
P(a < X < b) = P((a - μ) / σ < Z < (b - μ) / σ)

Theorems

Empirical Rule
68-95-99.7 Rule

Suitable Grade Level

Undergraduate / Advanced High School