b:head_first_statistics:estimating_populations_and_samples
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b:head_first_statistics:estimating_populations_and_samples [2019/12/10 13:37] – [How many gumballs? -- Probability of sample means] hkimscil | b:head_first_statistics:estimating_populations_and_samples [2022/11/15 23:44] – [Sampling distribution of sample mean] hkimscil | ||
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</ | </ | ||
- | \begin{eqnarray*} | + | \begin{align*} |
- | \overline{X} & = & \frac {\sum{X}}{n} \\ | + | \overline{X} & = \frac {\sum{X}}{n} \\ |
- | & = & \frac {\sum_{i=1}^{n} X_{i}}{n} \\ | + | & = \frac{ \sum_{i=1}^{n} X_{i} } {n} \\ |
- | & = & \hat{\mu} | + | & = \hat{\mu} |
- | \end{eqnarray*} | + | \end{align*} |
====== Estimating population variance ====== | ====== Estimating population variance ====== | ||
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{{: | {{: | ||
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+ | |||
+ | [[:Why N-1]] | ||
< | < | ||
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이 때 $n = 100$일때 각각의 시도에서의 (trial) proportion 기대값은 ($\hat{P}$): | 이 때 $n = 100$일때 각각의 시도에서의 (trial) proportion 기대값은 ($\hat{P}$): | ||
- | \begin{eqnarray} | ||
- | \hat{P_{1}} & = & {E(X_{1})}/ | ||
- | \hat{P_{2}} & = & {E(X_{2})}/ | ||
- | \hat{P_{3}} & = & {E(X_{3})}/ | ||
- | \hat{P_{4}} & = & {E(X_{4})}/ | ||
- | \cdots & \cdots & \cdots | ||
- | \hat{P_{k}} & = & {E(X_{k})}/ | ||
- | \end{eqnarray} | ||
- | 즉, $X \sim B(n, p)$ 일 때, sample의 확률 $P_{s} = \displaytype \frac{X}{n}$를 따른다 (X = red gumball이 나온 갯수, n = sample 크기). | + | \begin{align*} |
+ | n = 100, \\ | ||
+ | \hat{P_{1}} & = \frac{X_{1}}{n} = 0.34, (X_{1} = 34) \\ | ||
+ | \hat{P_{2}} & = \frac{X_{2}}{n} = 0.43, (X_{2} = 43) \\ | ||
+ | \hat{P_{3}} & = \frac{X_{3}}{n} = 0.32, (X_{3} = 32) \\ | ||
+ | \hat{P_{4}} & = \frac{X_{4}}{n} = 0.42, (X_{4} = 42) \\ | ||
+ | \cdots \cdots \cdots \\ | ||
+ | \hat{P_{k}} & = \frac{X_{k}}{n} = 0.24, (X_{1} = 24) \\ | ||
+ | \end{align*} | ||
+ | |||
+ | 즉, $X \sim B(n, p)$ 일 때, sample의 확률 $P_{s} = \dfrac{X}{n}$를 따른다 ($X$ = red gumball이 나온 갯수, | ||
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q <- 1-p | q <- 1-p | ||
n <- 100 | n <- 100 | ||
- | var <- (p*q)/(n-1) | + | var <- (p*q)/(n) |
- | se <- sqrt((p*q)/ | + | se <- sqrt((p*q)/ |
- | pnorm(.395, p, se, lower.tail = F) | + | o <- .4 |
+ | o.c <- .4 - (1/(2*n)) | ||
+ | o.c | ||
+ | pnorm(o.c, p, se, lower.tail = F) | ||
</ | </ | ||
< | < | ||
+ | > | ||
> p <- 0.25 | > p <- 0.25 | ||
> q <- 1-p | > q <- 1-p | ||
> n <- 100 | > n <- 100 | ||
- | > var <- (p*q)/(n-1) | + | > var <- (p*q)/(n) |
- | > se <- sqrt((p*q)/ | + | > se <- sqrt((p*q)/ |
- | > pnorm(.395, p, se, lower.tail = F) | + | > o <- .4 |
- | [1] 0.0004313594 | + | > o.c <- .4 - (1/(2*n)) |
+ | > o.c | ||
+ | [1] 0.395 | ||
+ | > pnorm(o.c, p, se, lower.tail = F) | ||
+ | [1] 0.0004060586 | ||
</ | </ | ||
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\overline{X} = \frac{X_{1} + X_{2} + . . . + X_{n}}{n} | \overline{X} = \frac{X_{1} + X_{2} + . . . + X_{n}}{n} | ||
\end{eqnarray*} | \end{eqnarray*} | ||
+ | 위는 풍선검 봉지 30개로 이루어진 샘플의 평균을 이야기하고 | ||
+ | 아래는 이 평균을 계속 모았을 때의 평균을 이야기한다. | ||
\begin{eqnarray*} | \begin{eqnarray*} | ||
E(\overline{X}) & = & E\left(\frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) | E(\overline{X}) & = & E\left(\frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) | ||
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\end{eqnarray*} | \end{eqnarray*} | ||
- | \begin{eqnarray*} | + | \begin{align*} |
- | Var(\overline{X}) & = & Var \left(\frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) \\ | + | Var(\overline{X}) & = Var \left(\frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) \\ |
- | & = & \frac{1}{n^2} | + | & = \frac {1}{n^2} Var \left(X_{1} + X_{2} + . . . + X_{n} \right) \\ |
- | & = & \frac{1}{n^2} | + | & = \frac{1}{n^2} (\sigma^2 + \sigma^2 + . . . + \sigma^2) \\ |
- | & = & \frac{1}{n^2} | + | & = \frac{1}{n^2} n * (\sigma^2) \\ |
- | & = & \frac{\sigma^2}{n} | + | & = \frac{\sigma^2}{n} |
- | \end{eqnarray*} | + | |
+ | |||
+ | \end{align*} | ||
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\end{eqnarray} | \end{eqnarray} | ||
- | $$\text{standard error of the sample means} = \frac{\sigma}{\sqrt{n}}$$ | + | \begin{eqnarray*} |
+ | \text{standard error} & = & \text{standard deviation | ||
+ | & = & \frac{\sigma}{\sqrt{n}} | ||
+ | & = & \sqrt{\frac{\sigma^{2}}{n}} | ||
+ | \end{eqnarray*} | ||
{{: | {{: |
b/head_first_statistics/estimating_populations_and_samples.txt · Last modified: 2022/11/17 12:47 by hkimscil