Confidence Interval - Mean, Proportion, Standard Deviation

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CI - Mean


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:

  • Use the function under Stat>Basic Statistics>1 Sample t.
  • Select the Confidence interval option and enter a confidence level, usually this level is 95% (default setting).

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.

CI - Proportion


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.

CI – Standard Deviation


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:

  • Use the function under Stat>Basic Statistics>Descriptive Statistics>Graphs>Graphical Summary.
  • Select the Graphs and Graphical Summary options. Enter a confidence level, usually this level is 95%.

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