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What is a confidence interval (CI)?

The point estimate provides only a single value estimate for a population parameter. It does not provide any information on how “good” the estimate is, or how close it is to the real value of the parameter.

A confidence interval provides an interval along with a certain confidence probability value (1) for a population parameter. A CI indicates the percentage of the CIs formed from a number of different samples of same size that would contain the real value of the estimated population parameter.

For example:

Let the the 95% confidence interval for the mean product length, calculated using a certain sample, is [24,29]. This means that if we kept taking similar samples to which we calculated the above CI from, about 95% of the CIs that we form will contain the real value of the mean length. Also, since (1) = 0.95, = 0.05

Types of CIs

- There are three types of CIs:
- Two-sided CI L U
- Lower one-sided CI L
- Upper one-sided CI - U
- There are three types of continuous random variables.
- Nominal-the-best (NTB) L U
- Larger-the-best (LTB) L
- Smaller-the-best (STB) - U
- Most common CIs are 95%, 99%, and 90% CIs.

Some Examples

- Nominal-the-better
- Clearance, chemical content (pH level), etc.
- We would like the value to be between two comsumer specification limits (LSL = m - and USL = m + ).
- Larger-the-best
- Strength, lifetime, reliability, etc.
- We would like the value to be larger than a consumer lower specification limit LSL only.
- Smaller-the-better
- Amount of error, time delay, monetary loss, etc.
- We would like the value to be smaller than a consumer upper specification limit USL only.

In general, to develop a parametric CI for a parameter , the sampling distribution (SMD) of its point estimator must be known and then used appropriately.

- Further, to obtain a lower one-sided CI, the value of should generally be placed at the upper tail of a distribution and vice a versa for an upper one-sided CI.

CONFIDENCE INTERVAL FOR THE MEAN OF A NORMAL PROCESS

WHEN THE VARIANCE IS KNOWN

CONFIDENCE INTERVAL FOR THE MEAN OF A NORMAL PROCESS

WHEN THE VARIANCE IS UNKNOWN

- In the previous example, the population variance was assumed to be known. However, in reality it is very likely that the real value of the variance will not be known.
- As the central limit theorem states, the sample average has approximately a normal distribution, whatever the parent distribution is, and:

- However, since the value of is not known, we will use the unbiased estimator S instead of . The standardized variable

- follows approximately the standard normal distribution only if n > 40. Therefore, we will use the standard normal distribution only if n > 40, but not if n < 40.

For small sample sizes, or if n<40:

- The standardized variable will be denoted by T;

follows a t (Student’s t) distribution with (n-1) degrees of freedom (df). The df is denoted by .

- The t distribution is always more spread out than the standard normal distribution, which accounts for the added variability due to the uncertainty about the real value of the population variance.
- For n > 40, the t distribution becomes practically equal to the standard normal distribution.

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