b:head_first_statistics:using_the_normal_distribution
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| b:head_first_statistics:using_the_normal_distribution [2025/10/08 12:02] – [X - Y] hkimscil | b:head_first_statistics:using_the_normal_distribution [2025/10/09 06:26] (current) – [X - Y] hkimscil | ||
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| rnorm(n, mean = 0, sd = 1) | rnorm(n, mean = 0, sd = 1) | ||
| </ | </ | ||
| - | </ | ||
| - | [{{ : | ||
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| + | {{: | ||
| 따라서 | 따라서 | ||
| $$P(X + Y < 380) = 0.9082409 $$ | $$P(X + Y < 380) = 0.9082409 $$ | ||
| + | </ | ||
| ===== exercise ===== | ===== exercise ===== | ||
| - | < | + | < |
| Julie’s matchmaker is at it again. What's the **probability that a man will be at least 5 inches taller than a woman**? In Statsville, the height of men in inches is distributed as N(71, 20.25), and the height of women in inches is distributed as N(64, 16). | Julie’s matchmaker is at it again. What's the **probability that a man will be at least 5 inches taller than a woman**? In Statsville, the height of men in inches is distributed as N(71, 20.25), and the height of women in inches is distributed as N(64, 16). | ||
| </ | </ | ||
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| **probability that a man will be at least 5 inches taller than a woman**? = " | **probability that a man will be at least 5 inches taller than a woman**? = " | ||
| + | |||
| \begin{align*} | \begin{align*} | ||
| P(X > F + 5) & = P(X - F > 5) | P(X > F + 5) & = P(X - F > 5) | ||
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| {{: | {{: | ||
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| Q: So what’s the difference between linear transforms and independent observations? | Q: So what’s the difference between linear transforms and independent observations? | ||
| A: Linear transforms affect the underlying values in your probability distribution. As an example, if you have a length of rope of a particular length, then applying a linear transform affects the length of the rope. Independent observations have to do with the quantity of things you’re dealing with. As an example, if you have n independent observations of a piece of rope, then you’re talking about n pieces of rope. In general, __if the quantity changes__, you’re dealing with **independent observations**. __If the underlying values change__, then you’re dealing with a **transform**. | A: Linear transforms affect the underlying values in your probability distribution. As an example, if you have a length of rope of a particular length, then applying a linear transform affects the length of the rope. Independent observations have to do with the quantity of things you’re dealing with. As an example, if you have n independent observations of a piece of rope, then you’re talking about n pieces of rope. In general, __if the quantity changes__, you’re dealing with **independent observations**. __If the underlying values change__, then you’re dealing with a **transform**. | ||
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| [1] 0.9452007 | [1] 0.9452007 | ||
| # 혹은 | # 혹은 | ||
| - | > pnorm(800, 720, sqrt(2500), lower.tail = TRUE) | + | > pnorm(800, 720, sqrt(2500), |
| + | > lower.tail = TRUE) | ||
| [1] 0.9452007 | [1] 0.9452007 | ||
| </ | </ | ||
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| Before going further: | Before going further: | ||
| - | < | + | < |
| So what’s the probability of getting 30 or more questions right out of 40? That will help us determine whether to keep playing, or walk away. | So what’s the probability of getting 30 or more questions right out of 40? That will help us determine whether to keep playing, or walk away. | ||
| </ | </ | ||
| - | < | + | < |
| There are 40 questions, which means there are 40 trials. | There are 40 questions, which means there are 40 trials. | ||
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| </ | </ | ||
| - | < | + | < |
| < | < | ||
| > pbinom(29, | > pbinom(29, | ||
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| {{: | {{: | ||
| - | < | + | < |
| 이를 R을 이용하여 구하면, | 이를 R을 이용하여 구하면, | ||
| < | < | ||
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| 이 값은 위의 0.387에 근사하다. | 이 값은 위의 0.387에 근사하다. | ||
| - | < | + | < |
| * In particular circumstances you can **use the normal distribution to approximate the binomial**. If X ~ B(n, p) and np > 5 and nq > 5 then you can approximate X using X ~ N(np, npq) | * In particular circumstances you can **use the normal distribution to approximate the binomial**. If X ~ B(n, p) and np > 5 and nq > 5 then you can approximate X using X ~ N(np, npq) | ||
| * If you’re approximating the binomial distribution with the normal distribution, | * If you’re approximating the binomial distribution with the normal distribution, | ||
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| {{: | {{: | ||
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| Q:Does it really save time to approximate the binomial distribution with the normal? | Q:Does it really save time to approximate the binomial distribution with the normal? | ||
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| ===== Pool Puzzle ===== | ===== Pool Puzzle ===== | ||
| <wrap # | <wrap # | ||
| - | < | + | < |
| X < 3 ---- <wrap spoiler> X < 2.5 </ | X < 3 ---- <wrap spoiler> X < 2.5 </ | ||
| X > 3 ---- <wrap spoiler> X > 3.5 </ | X > 3 ---- <wrap spoiler> X > 3.5 </ | ||
b/head_first_statistics/using_the_normal_distribution.1759892568.txt.gz · Last modified: by hkimscil