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Lecture 14. Chapter 7 Statistical Intervals Based on a Single Sample. What is a confidence interval (CI)?

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lecture 14

Lecture 14

Chapter 7

Statistical Intervals Based on a Single Sample

slide2
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

slide3
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.
slide4
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.
slide5
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.
slide9
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.
slide10
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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