Textbook Question
Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.
c = 0.98
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7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
1:01 minutesProblem 5.4.5
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 mean of the distribution of sample means increases.
Verified step by step guidance1
Understand the concept of the sampling distribution of the sample mean: The sampling distribution of the sample mean is the probability distribution of all possible sample means from a population. Its mean is equal to the population mean (denoted as μ).
Recall the property of the mean of the sampling distribution: The mean of the sampling distribution of the sample mean does not depend on the sample size. It is always equal to the population mean (μ).
Analyze the statement: The statement claims that as the sample size increases, the mean of the distribution of sample means increases. This is incorrect because the mean of the sampling distribution remains constant and equal to μ, regardless of the sample size.
Rewrite the statement as a true statement: 'As the sample size increases, the variability (standard deviation) of the distribution of sample means decreases, but the mean of the distribution of sample means remains constant and equal to the population mean.'
Conclude: The original statement is false, and the corrected true statement has been provided in the previous step.
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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 as the sample size increases, the distribution of the sample means approaches a normal distribution, regardless of the shape of the population distribution. This theorem is fundamental in statistics as it allows for the use of normal probability techniques for inference, even when the original data is not normally distributed.
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Calculating the Mean
Sample Mean
The sample mean is the average of a set of values taken from a larger population. It serves as an estimate of the population mean and is calculated by summing all sample values and dividing by the number of observations. Understanding how the sample mean behaves as sample size increases is crucial for making inferences about the population.
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Law of Large Numbers
The Law of Large Numbers states that as the size of a sample increases, the sample mean will converge to the expected value (population mean). This principle underlines the importance of larger sample sizes in achieving more accurate and reliable estimates, reinforcing the idea that variability decreases with larger samples.
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