Textbook Question
Normal Quantile Plot The accompanying normal quantile plot was obtained from the longevity times of presidents. What does this graph tell us?
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7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
1:45 minutesProblem 5.4.6
Textbook Question
True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
As the sample size increases, the standard deviation of the distribution of sample means increases.
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Understand the concept of the standard deviation of the distribution of sample means, also known as the standard error. It is calculated as the population standard deviation divided by the square root of the sample size: , where is the population standard deviation and is the sample size.
Analyze the relationship between the sample size () and the standard error. As increases, the denominator of the formula increases, which causes the standard error to decrease.
Evaluate the given statement: 'As the sample size increases, the standard deviation of the distribution of sample means increases.' Based on the formula and the relationship, this statement is false because the standard error actually decreases as the sample size increases.
Rewrite the statement to make it true: 'As the sample size increases, the standard deviation of the distribution of sample means decreases.'
Conclude that the correct understanding of the relationship between sample size and the standard error is essential for interpreting the behavior of sampling distributions.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In statistics, it quantifies how much individual data points differ from the mean of the dataset. A smaller standard deviation indicates that the data points tend to be close to the mean, while a larger standard deviation indicates more spread out values.
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Central Limit Theorem
The Central Limit Theorem states that, as the sample size increases, the distribution of the sample means will approach a normal distribution, regardless of the shape of the population distribution. This theorem is fundamental in statistics because it allows for the use of normal probability techniques for inference, even when the original data is not normally distributed.
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Distribution of Sample Means
The distribution of sample means, also known as the sampling distribution, is the probability distribution of all possible sample means from a population. As the sample size increases, the standard deviation of this distribution, known as the standard error, decreases, leading to more precise estimates of the population mean. This concept is crucial for understanding how sample size affects statistical inference.
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