Ch 6.7 - 136MEE
Chapter 6, Problem 136MEE(choose chapter or problem)
Suppose that you have a sample \(x_{1}, x_{2}, \ldots, x_{n}\) and have calculated \(\bar{x_{n}}\) and \(S_{n}^{2}\) for the sample. Now an \((n+1)\) st observation becomes available. Let \(\bar{x}_{n+1}\) and \(S_{n+1}^{2}\) be the sample mean and sample variance for the sample using all \(n + 1\) observations.
(a) Show how \(S_{n+1}^{2}\) can be computed using \(\bar{x_{n}}\) and \(\bar{x}_{n+1}\)
(b) \(n s_{n+1}^{2}=(n-1) S_{n}^{2}+\frac{n\left(x_{n+1}-\bar{x_{n}}\right)^{2}}{n+1}\)
(c) Use the results of parts (a) and (b) to calculate the new sample average and standard deviation for the data of Exercise 6-38, when the new observation is \(x_{38}=64\).
Equation Transcription:
Text Transcription:
x_{1}, x_{2}, ..., x_{n}
bar{x_{n}}
S_{n}^{2}
(n + 1)
bar{x}_{n+1}
S_{n+1}^{2}
n + 1
S_{n+1}^{2}
bar{x_{n}}
bar{x}_{n+1}
n s_{n+1}^{2}=(n-1) S_{n}^{2} + frac{n (x_{n+1} - bar{x_{n}})^{2}}{n+1}
x_{38}=64
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