MGSC 312 Lecture Notes - Lecture 8: Confidence Interval, Interval Estimation, Normal Distribution
19 May 2018
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Chapter 8
• Population mean
o In quality control ap plications where a process is assumed to be operating correctly, or
i otrol, it is appro- priate to treat the population standard deviation as known.
o This is called a known case.
• Margin of Error and the Interval Estimate
o Using the standard normal probability table, we find that 95% of the values of any nor-
mally distributed random variable are within 1.96 standard deviations of the mean.
Thus, when the sampling distribution of x ̄ is normally distributed, 95% of the x ̄ values
must be within 1.96σx ̄ of the mean μ. In the Lloyd’s example we know that the
sampling distribu- tion of x ̄ is normally distributed with a standard error of σx ̄ 2.
Because 1.96σx ̄ 1.96(2) 3.92, we can conclude that 95% of all x ̄ values obtained
using a sample size of n 100 will be within 3.92 of the population mean μ.
• Confidence interval
o In a normal distribution if you are 2 sd’s aa fro the ea ou are 95% ofidet
that the intervals constructed will contain the population mean we say that we are 95%
confident that the interval specified contains the population mean
o 95% here is the confidence level
o 0.95 is the confidence coefficient
o The interval specified is the confidence interval
• Population Mean : population standard deviation
o When developing an interval estimate of a population mean we usually do not have a
good estimate of the population standard deviation either. In these cases, we must use
the same sample to estimate both μ ad σ.
o Whe s is used to estiate σ, the argi of error ad the iteral estiate for the
population mean are based on the t distribution
o The t distribution is a family of similar probability distributions, with a specific t dis-
tribution depending on a parameter known as the degrees of freedom.
o As the number of degrees of freedom increases, the difference between the t
distribution and the standard normal distribution becomes smaller and smaller
o t distribution with more degrees of freedom exhibits less variability and more closely
resembles the standard normal distribution.
o The mean of the t dis- tribution is zero.
o If the degrees of freedom exceed 100, the infinite degrees of freedom row can be used
to approximate the actual t value
• Margin of error and interval estimate
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