목차

• Sampling Distributions
• Sampling Distributions of the Proportion
  • Example
• 문제

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Sampling Distributions

표본분포이다.
• 표본 분포는 모집단에서 선택한 표본 크기에 대해 표본 통계량의 가능한 모든 값의 분포이다.

• 샘플링을 여러번 하여 여러번 한 샘플의 분포를 측정하여 이들을 다 종합했을 떄 보이는 그림
• distribution of all of the possible values of a sample statistic for a given sample size selected from a population.

Sampling Distributions of the Proportion

이것은 확률의 sample을 계속 뽑아서 그것의 확률을 계속 구해 이 확률로 히스토그램을 만든다고 생각하면 된다.

이해하기 어렵다면 Sampling Distributions of the Mean을 보고오자!

여기서 smapling 후 mean을 구하는데 하닌 확률을 구한다는 것만 다르다.

Sampling Distributions of the Proportion은 Sampling Distributions of the Mean과 비교해 딱 2가지만 달라진다.

1️⃣
population은 standard deviation이 없다.
➡️ 상식적으로 확률엔 표준 분포가 존재할수 없다 확률 자체니까

그런데 이를 뽑아서 모아둔 히스토그램에는 standard deviation이 있음에 유의하자

그래서 mean과 달리 위와 같은 식이 적용된다

2️⃣
또한 Central Limit Theorem에 의해 n>30 이였던 mean과 달리

proportion은 아래와 같은 조건만 만족시키면 된다

즉, 그러므로

[Z-Value for Proportions]

Example

If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?

if π = 0.4 and n = 200,
what is P(0.40 ≤ p ≤ 0.45) ?

⛔ Proposition과 probability를 헷갈리지 말자!

문제

[1 - 3]
The amount of pyridoxine (in grams) per multiple vitamin is normally distributed with μ = 110 grams and σ = 25 grams. A sample of 25 vitamins is to be selected.

1. What is the probability that the sample mean will be less than 100 grams?

2. So, 95% of all sample means will be greater than how many grams?

3. So, the middle 70% of all sample means will fall between what two values?

4. The Central Limit Theorem is important in statistics because
a) for a large n, it says the population is approximately normal.
b) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size.
c) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
d) for any sized sample, it says the sampling distribution of the sample mean is approximately normal.

5. The standard error of the mean for a sample of 100 is 30. In order to cut the standard error of the mean to 15, we would
a) increase the sample size to 200.
b) increase the sample size to 400.
c) decrease the sample size to 50.
d) decrease the sample to 25.

[6 - 8]
According to an article, 19% of the entire U.S. population have high-speed access to the NETFLIX. Random samples of size 200 are selected from the U.S. population.

6. Among all the random samples of size 200, ______ % will have between 9% and 29% who have high-speed access to the NETFLIX.

1. Among all the random samples of size 200, __ % will have more than 30% who have high-speed access to the NETFLIX.
2. Among all the random samples of size 200, __ % will have less than 20% who have high-speed access to the NETFLIX.
3. Sales prices of baseball cards from the 1960s are known to possess a skewed-right distribution with a mean sale price of $5.25 and a standard deviation of $2.80. Suppose a random sample of 100 cards from the 1960s is selected. Describe the sampling distribution for the sample mean sale price of the selected cards. a) Skewed-right with a mean of $5.25 and a standard error of $2.80 b) Normal with a mean of $5.25 and a standard error of $0.28 c) Skewed-right with a mean of $5.25 and a standard error of $0.28 d) Normal with a mean of $5.25 and a standard error of $2.80
4. Major league baseball salaries averaged $1.5 million with a standard deviation of $0.8 million in
5. Suppose a sample of 100 major league players was taken. Find the approximate probability that the average salary of the 100 players exceeded $1 million. a) Approximately 0 b) 0.2357 c) 0.7357 d) Approximately 1
6. At a computer manufacturing company, the actual size of computer chips is normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A random sample of 12 computer chips is taken. What is the standard error for the sample mean? a) 0.029 b) 0.050 c) 0.091 d) 0.120