6.13
표본 분포
각 표본에 대한 평균, 비율 또는 표준 편차와 같은 측정된 특성을 가진 주어진 모집단에서 크기 n의 단순 무작위 표본이 주어지면 측정된 모든 특성의 확률 분포를 표본 추출 분포라고 합니다. 통계가 표본마다 달라지는 정도를 통계의 표본 변동성이라고 합니다. 일반적으로 표…
10개의 피커 휠을 돌리고 결과의 평균을 찾는 것을 고려하십시오. 이 과정은 20,000번 반복됩니다.
프로세스의 각 반복에 대해 얻은 표본 평균이 표시되며, 이는 정규 분포 그래프와 유사합니다.
표본 크기가 크면 분포가 정규 분포에 가까워지고 표본 평균의 평균이 모평균에 가까워집니다.
평균, 분산 또는 표본 비율과 같은 통계량의 값 분포를 표본 추출 분포라고 합니다.
평균과 마찬가지로 각 표본에 대한 분산을 구하고 오른쪽으로 치우쳐 보이는 빈도 분포를 그릴 수 있습니다.
이 경우에도 표본 크기가 크면 표본 분산의 평균이 모집단 분산에 가깝습니다.
각 표본에서 홀수의 비율을 고려하고 그래프를 그리면 분포는 대략적으로 정규 분포 패턴을 따릅니다.
평균 및 분산과 유사하게, 표본 크기가 크면 표본 비율의 평균이 모집단 비율에 가깝습니다.
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Q1: What is a sampling distribution and how is it created?
A sampling distribution is the probability distribution of a statistic, such as the mean, variance, or proportion, calculated from multiple simple random samples of the same size from a population. It is created by repeatedly drawing samples, computing the statistic for each sample, and plotting the frequency distribution of those statistics.
Q2: How does sample size affect the shape of a sampling distribution?
As sample size increases, the sampling distribution approaches a normal distribution shape, regardless of the original population distribution. This convergence means that larger samples produce more reliable estimates, with the mean of sample statistics becoming increasingly close to the true population parameter.
Q3: What is sampling variability and how is it measured?
Sampling variability refers to how much a statistic varies from one sample to another. It is measured using the standard error, which is the standard deviation of the sampling distribution. The standard error of the mean is a common example that quantifies the variability of sample means around the population mean.
Q4: Why does the mean of sample means approach the population mean?
When sample size is large, the mean of sample means converges to the population mean due to the law of large numbers. This property holds for other statistics as well: the mean of sample variances approaches the population variance, and the mean of sample proportions approaches the population proportion.
Q5: Can sampling distributions be used for different types of statistics?
Yes, sampling distributions can be constructed for any statistic, including the mean, variance, and proportion. Each statistic has its own sampling distribution with distinct characteristics. For example, the distribution of sample proportions typically follows an approximately normal pattern when sample size is sufficiently large.
Q6: What role does standard error play in statistical inference?
Standard error measures the precision of a sample statistic as an estimate of the population parameter. A smaller standard error indicates that sample statistics cluster more tightly around the true population value, making the estimate more reliable for drawing conclusions about the population.
Q7: How do sampling distributions relate to probability distributions?
A sampling distribution is a specific type of probability distribution that describes the likelihood of different values for a sample statistic. While probability distributions characterize outcomes of random variables, sampling distributions characterize the behavior of statistics computed from repeated samples.