### Understanding Confidence Intervals in ManufacturingThe production process for engine control housing units of a particular type has recently been modified. Initially, it was known that the distribution of hole diameters for bushings on these housings was normal, with a standard deviation of 0.100 mm. After modification, it's assumed that while the distribution shape and standard deviation remain unchanged, the mean diameter might have shifted. A sample of 40 housing units is examined to determine hole diameter, resulting in a sample mean diameter of 5.426 mm.#### Calculating the Confidence IntervalTo find a confidence interval for the true average hole diameter with a confidence level of 90%, we follow these steps:1. **Determine the Confidence Level**: $100(1 - \alpha) = 90\%$, thus $\alpha = 0.10$.2. **Z-Value**: The z-value for $\alpha/2$ (0.05) is $z_{0.05} = 1.645$, which corresponds to a cumulative z-curve area of 0.9500.3. **Confidence Interval Calculation**: \[ 5.426 \pm \left(1.645 \times \frac{0.100}{\sqrt{40}}\right) = 5.426 \pm 0.026 = (5.400, 5.452) \]The desired interval thus ranges from 5.400 mm to 5.452 mm.#### ConclusionWith reasonably high confidence, it can be said that the true mean diameter $\mu$ lies within the interval 5.400 mm to 5.452 mm. The interval is narrow, reflecting the small variability in hole diameter ($\sigma = 0.100$). This indicates precision in manufacturing despite potential changes in the mean.

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The production process for engine control housing units of a particular type has recently been modified. Prior to this modification, historical data had suggested that the distribution of hole diameters for bushings on the housings was normal with a standard deviation of 0.100 mm. It is believed that the modification has not affected the shape of the distribution or the standard deviation, but that the value of the mean diameter may have changed. A sample of 40 housing units is selected and hole diameter is determined for each one, resulting in a sample mean diameter of 5.426 mm. Let's calculate a confidence interval for true average hole diameter using confidence level of 90%. This requires that 100(1 - a) = , from which a = and z/2 - Z0.05 - 1.645 (corresponding to a cumulative z-curve area of 0.9500). The desired interval is then 0.100 5.426 t = 5.426 + 0.026 = (5.400, 5.452) V40 With a reasonably high degree of confidence, we can say that 5.400 < u < 5.452. This interval is rather narrow because of the small amount of variability in hole diameter (o = 0.100).

The production process for engine control housing units of a particular type has recently been modified. Prior to this modification, historical data had suggested that the distribution of hole diameters for bushings on the housings was normal with a standard deviation of 0.100 mm. It is believed that the modification has not affected the shape of the distribution or the standard deviation, but that the value of the mean diameter may have changed. A sample of 40 housing units is selected and hole diameter is determined for each one, resulting in a sample mean diameter of 5.426 mm. Let's calculate a confidence interval for true average hole diameter using confidence level of 90%. This requires that 100(1 - a) = , from which a = and z/2 - Z0.05 - 1.645 (corresponding to a cumulative z-curve area of 0.9500). The desired interval is then 0.100 5.426 t = 5.426 + 0.026 = (5.400, 5.452) V40 With a reasonably high degree of confidence, we can say that 5.400 < u < 5.452. This interval is rather narrow because of the small amount of variability in hole diameter (o = 0.100).

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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