Confidence Intervals
Applied Example where σ is Unknown
Learning Objectives
In this section, we will do the following calculations to estimate the true mean when the population standard deviation (σ) is unknown:
- Review an applied example where confidence intervals are well suited
- Compare to the previous example where confidence intervals were not well suited
- Calculate the lower and upper limits of the confidence interval
- Calculate the required sample size given a required maximum margin of error
In the previous section, we purposefully used an example that did poorly at predicting future demand. In contrast, in this section, we will choose an example that is well suited to using confidence intervals to analyze the quality control on the product(s) produced.
Quality Control on Space Fasteners
We will look at quality control for space fasteners that could be supplied to organizations and companies like NASA, SpaceX and other companies that build satellites and other machines used in space. These companies must have very strict quality control and adherence to requirements policies.
 |
 |
Space fasteners, which would be machine-produced should be fairly uniform and doing analysis based off of their sample statistics such as mean and standard deviation is very appropriate. The parts should have little variation in dimensions and standard deviations would be a great measure to determine the level of variation in the the parts’ dimensions, materials’ performance and so on.
Constructing a Confidence Interval using T.INV.2T (EXAMPLE)
We will produce confidence intervals for several types of space fasteners in the examples below. In this first example, we will use Excel’s T.INV.2T() function.
Example 51.1
Problem Setup: A company supplies Standard Hexagon Head Cap Screws. They perform regular quality control checks by sampling some of the screws produced and measuring and testing the screws:
- They sample 100 screws.
- The mean length of the screws sampled is 0.6249 inches.
- The standard deviation of the lengths is 0.013 inches.
Question: Construct the 99% confidence interval for the screw lengths. What are the highest and lowest average screw length with 99% certainty?
Solution: Click here to download the Excel file shown in the video below.
asdf
Conclusion: There is a 99% chance that the average bolt length is between 0.6215 and 0.6283 inches in length.
Click here to reveal the written solutions from the above video
Solution: Let us first write out the values needed for T.INV.2T:
- [latex]n = 100[/latex]
- [latex]df = n - 1 = 100 - 1 = 99[/latex]
- [latex]\alpha = 1-0.99 = 0.01[/latex]
This gives: [latex]t = \text{T.INV.2T}(\alpha, df) = \text{T.INV.2T}(0.01, 99) = 2.6264[/latex]
To build the confidence interval, we use the [latex]t[/latex]-score above and the following values:
- [latex]\bar{x} = 0.6249[/latex]
- [latex]s = 0.013[/latex]
This gives:
- [latex]CL_{Lower} = \bar{x}-t \cdot \frac{s}{\sqrt{n}} = 0.6249 - 2.6264 \cdot \frac{0.013}{\sqrt{100}} = 0.6249 - 0.003414 = 0.6215[/latex]
- [latex]CL_{Upper} = \bar{x}+t \cdot \frac{s}{\sqrt{n}} = 0.6249 + 2.6264 \cdot \frac{0.013}{\sqrt{100}} = 0.6249 + 0.003414 = 0.6283[/latex]
Conclusion: There is a 99% chance that the average bolt length is between 0.6215 and 0.6283 inches in length.
Constructing a Confidence Interval using T.INV (Video)
Let us next look at an example where quality control is performed on shear bolts. In this example, we will use Excel’s T.INV() to calculate the [latex]t[/latex]-score.
Example 51.2
Problem Setup: An large order of shear bolts has been received at a spacecraft manufacturing company.
They perform quality control on the order received before paying the manufacturer for the order. They find the following sample statistics based off of the 400 shear bolts they sampled:
- The average bolt length is 0.532 inches
- The standard deviation of the bolt lengths is 0.010 inches.
The quality control team constructs the 90% confidence interval for the bolt lengths based off of the sample statistics they found.
Question: What confidence interval limits would the team have calculated?
asfd
Solution: Click here to download the Excel file shown in the video below.
Conclusion:
Click here to reveal the solutions shown in the video above.
asdf
Constructing a Confidence Interval using CONFIDENCE.T (Video)
asdf
Example 49.1.3 Using CONFIDENCE.T()
asdf
Click here to reveal the written solutions from the above video
asfd
Calculating the Required Sample Size (Video)
asdf
Example 49.1.2
adsf
Click here to reveal the written solutions from the above video
asdf
