b:head_first_statistics:estimating_populations_and_samples
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| b:head_first_statistics:estimating_populations_and_samples [2025/11/04 22:27] – [Sampling distribution of sample mean] hkimscil | b:head_first_statistics:estimating_populations_and_samples [2025/11/04 23:18] (current) – [Exercise] hkimscil | ||
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| | mean of $\overline{X}s = E(\overline{X}) $ | $ E(\overline{X}) = \displaystyle E\left( \frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) $ |||||| | | mean of $\overline{X}s = E(\overline{X}) $ | $ E(\overline{X}) = \displaystyle E\left( \frac{X_{1} + X_{2} + . . . + X_{n}}{n}\right) $ |||||| | ||
| + | < | ||
| + | * b: | ||
| + | * b: | ||
| + | </ | ||
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| ===== Using CLT for the binomial distribution ===== | ===== Using CLT for the binomial distribution ===== | ||
| $X \sim B(n, p)$ 에서 $\mu = np$, $\sigma^2 = npq$ 이고, | $X \sim B(n, p)$ 에서 $\mu = np$, $\sigma^2 = npq$ 이고, | ||
| - | n이 30이 넘는 조건에서 이항분포가 정상분포를 이룬다고 하므로 | + | n이 30이 넘는 조건에서 이항분포가 정상분포를 이룬다고 하므로 |
| - | $\overline{X} \sim N(\mu, \frac{\sigma^2}{n})$에 대입해 보면: | + | $\overline{X} \sim N (\mu, \frac{\sigma^2}{n} ) $에 대입해 보면: |
| - | $$\overline{X} \sim N(np, \; pq) $$ | + | $$\overline{X} \sim N \left(np, \; pq \right) $$ |
| {{: | {{: | ||
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| > pnorm(b/a) | > pnorm(b/a) | ||
| [1] 1.053435e-16 | [1] 1.053435e-16 | ||
| - | |||
| </ | </ | ||
| + | |||
| + | |||
| ====== Recap ====== | ====== Recap ====== | ||
| Distribution of **Sample** <fc # | Distribution of **Sample** <fc # | ||
b/head_first_statistics/estimating_populations_and_samples.1762295263.txt.gz · Last modified: by hkimscil