8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
1:53 minutesProblem 6.3.6a
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.
a. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)?
Verified step by step guidance1
The problem involves the distribution of sample means, which is a key concept in the Central Limit Theorem (CLT). According to the CLT, regardless of the shape of the population distribution (in this case, skewed), the distribution of the sample means will approach a normal distribution as the sample size increases.
The sample size in this problem is 40, which is considered sufficiently large for the Central Limit Theorem to apply. This means the distribution of the sample means will be approximately normal.
The Central Limit Theorem applies because the sample size (n = 40) is greater than 30, which is a common threshold for the theorem to hold true in practice.
The approximate shape of the distribution of the sample means is therefore normal, even though the original population distribution of annual incomes is skewed.
This result is important because it allows us to use statistical methods that assume normality (e.g., z-scores, confidence intervals) when analyzing the sample means.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Central Limit Theorem
The Central Limit Theorem states that the distribution of the sample means will tend to be normal, regardless of the shape of the population distribution, as long as the sample size is sufficiently large (typically n ≥ 30). This means that even if the incomes of college presidents are skewed, the means of multiple samples will approximate a normal distribution.
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Sampling Distribution
A sampling distribution is the probability distribution of a statistic (like the mean) obtained from a large number of samples drawn from a specific population. In this case, the sampling distribution of the mean annual income of college presidents will show how the means vary from sample to sample, and its shape will be influenced by the Central Limit Theorem.
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Skewness
Skewness refers to the asymmetry in the distribution of data. A distribution is skewed if one tail is longer or fatter than the other. In the context of this question, the annual incomes of college presidents are skewed, but the means of samples taken from this population will still form a normal distribution due to the Central Limit Theorem.
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