8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
2:02 minutesProblem 6.3.6b
Textbook Question
College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.
b. What value do the sample means target? That is, what is the mean of all such sample means?
Verified step by step guidance1
Step 1: Understand the concept of the sampling distribution of the sample mean. When we take many random samples of size n (in this case, n = 40) from a population, the sample means form their own distribution, known as the sampling distribution of the sample mean.
Step 2: Recall the property of the sampling distribution of the sample mean. According to the Central Limit Theorem, the mean of the sampling distribution of the sample mean is equal to the mean of the population, regardless of whether the population distribution is skewed or normal.
Step 3: Identify the population mean. The problem does not explicitly provide the population mean, but the sample means will target this value. If the population mean is known, it can be directly used as the mean of the sampling distribution.
Step 4: Note that the shape of the population distribution (skewed in this case) does not affect the mean of the sampling distribution. The mean of the sample means will always target the population mean.
Step 5: Conclude that the value the sample means target is the population mean. If the population mean is not provided, it remains a theoretical value that the sample means are centered around.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution
The sampling distribution is the probability distribution of a statistic obtained by selecting random samples from a population. In this case, it refers to the distribution of the sample means of the annual incomes of college presidents. According to the Central Limit Theorem, regardless of the population's distribution, the sampling distribution of the sample means will tend to be normal if the sample size is sufficiently large.
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Mean of Sample Means
The mean of all sample means, also known as the expected value of the sample mean, is equal to the population mean. This principle states that if you take many samples from a population and calculate their means, the average of those means will converge to the true mean of the population, even if the population distribution is skewed.
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Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental statistical principle that states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the original population's distribution. This theorem is crucial for understanding how sample means behave and allows statisticians to make inferences about the population mean based on sample data.
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