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Purpose
A confidence interval is used to estimate the true mean of a continuous CT characteristic within a range of values based on observations from a sample. A pre-assigned probability called “confidence level” is used to determine this range of values. For a confidence level of 95%, we can say that we have a 95% percent chance that the CI contains the true mean of the CT characteristic.
Anatomy
Terminology
A. True or Population Mean.
B. Sample Mean.
C. Sample Standard Deviation.
D. Sample Size (n).
E. Tabulated Student (t) distribution value with alpha risk and n-1 degrees of freedom.
F. Lower Confidence Limit (LCL).
G. Upper Confidence Limit (UCL).
H. A representation of a confidence interval. For a confidence level of 95%, we can say that we have a 95% percent chance that the true mean is somewhere in the interval between the Lower Confidence Limit (LCL) and the Upper Confidence Limit (UCL).
Major Considerations
The assumption for using this interval is that the data comes from a normal distribution. The use of the Student (t) distribution is for sample sizes of less than 30. When the sample size is greater or equal to 30, the standard normal distribution (Z) is generally used. However, as the sample size increases, the t distribution approaches the Z distribution (see Tool Distribution – t).
Application Cookbook
1. Collect a sample of data from the process.
2. Analyze data with Minitab:
Note 1: If the sample size is ³ 30, the function under Stat>Basic Statistics>1 Sample Z can be used.
Note 2: There is another function that can be used to obtain the confidence interval for the mean under Stat>Basic Statistics>Descriptive Statistics>Graphs>Graphical Summary. This function presents not only this interval but also descriptive statistics and graphical representations.
Purpose
To calculate the confidence interval for a proportion which contains the true value of p with a (1 - alpha) level of confidence.
Anatomy
Reference: Business Statistics
Terminology
A. P-hat is the observed proportion in the sample and is an estimator of the population proportion.
B. p is the population proportion.
C. n is sample size.
D. Za/2 is the standard normal deviate appropriate to the level of confidence.
E. a is the chosen level of a risk.
F. Standard deviation for p-hat .
Major Considerations
The interval is based on an approximation of the Binomial distribution by a Normal Distribution.
Application Cookbook
1. Determine p-hat from sample data.
2. Decide on a (alpha risk), e.g. .05.
3. Determine Za/2 from tables or Excel, e.g. Za/2 = 1.96 for a(alpha) = .05.
4. Apply the formula.
Purpose
To estimate the true standard deviation of a continuous CT characteristic within a range of values based on observations from a sample. A pre-assigned probability called "confidence level" is used to determine this range of values. For a confidence level of 95%, we can say that we have a 95% percent chance that the confidence interval contains the true standard deviation of the CT characteristic.
Anatomy
Reference: The Vision of Six Sigma: Tools and Methods for Breakthrough by M. Harry
Terminology
A. True or Population Standard Deviation.
B. Sample Standard Deviation.
C. Sample Size (n).
D. Tabulated value of a Chi-square with risk and n-1 degrees of freedom.
E. Tabulated value of a Chi-square distribution with risk and n-1 degrees of freedom.
F. Lower Confidence Limit (LCL).
G. Upper Confidence Limit (UCL).
H. A representation of a confidence interval. For a confidence level of 95%, we can say that we have a 95% percent chance that the true standard deviation is somewhere in the interval between the Lower Confidence Limit (LCL) and the Upper Confidence Limit (UCL).
Major Considerations
The assumption for using this interval is that the data comes from a normal distribution.
Application Cookbook
1. Collect a sample of data from the process.
2. Use the following Minitab function. The function available presents not only this interval but also descriptive statistics and graphical representations:
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