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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