Boeing, Inc. is attempting to determine whether an existing
machine is capable of milling an engine part that has a design
specification of 5.0mm+- 0.10 mm. The standard deviation of the
process is
unknown, so a sample of 25 engine parts is selected and measured.
The sample mean is found to be 4.95mm, with a sample standard deviation of 0.08mm. Can the existing machine produce engine parts within specification? Use a 5% level of significance.
To determine whether the existing machine can produce engine parts within specification, we can use a one-sample t-test. The null hypothesis is that the true mean of the milling process is 5.0mm, and the alternative hypothesis is that it is different from 5.0mm. We can use a two-tailed test since we are not given a specific directionality in the alternative hypothesis. The test statistic is computed as follows:
t = (sample mean - hypothesized mean) / (sample standard error)
sample standard error = sample standard deviation / sqrt(sample size)
t = (4.95 - 5.0) / (0.08 / sqrt(25))
t = -1.25
Using a t-table with 24 degrees of freedom (since we have a sample size of 25 and are estimating one parameter), we find the critical t-value for a two-tailed test with a 5% level of significance to be approximately 2.064. Since our computed t-value of -1.25 is less than the critical t-value, we fail to reject the null hypothesis. There is not enough evidence to suggest that the true mean of the milling process is different from the design specification of 5.0mm, and we can conclude that the existing machine is capable of producing engine parts within specification.
To determine whether an existing machine is capable of milling an engine part, we need to calculate the capability index called Cp. Cp measures the potential capability of the process to meet the design specification.
The formula for Cp is:
Cp = (Upper Specification Limit - Lower Specification Limit) / (6 * Standard Deviation)
In this case, the design specification is 5.0mm +- 0.10 mm. This means the upper specification limit (USL) is 5.10 mm and the lower specification limit (LSL) is 4.90 mm.
We're given the standard deviation (σ) of the process.
Let's calculate Cp using the given values.