8. Sampling Distributions & Confidence Intervals: Proportion
Confidence Intervals for Population Proportion
1:09 minutesProblem 6.R.4b
Textbook Question
Arm Circumferences Arm circumferences of adult men are normally distributed with a mean of 33.64 cm and a standard deviation of 4.14 cm (based on Data Set 1 “Body Data†in Appendix B). A sample of 25 men is randomly selected and the mean of the arm circumferences is obtained.
b. What is the mean of all such sample means?
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Identify the key information provided in the problem: the population mean (μ = 33.64 cm), the population standard deviation (σ = 4.14 cm), and the sample size (n = 25).
Recall the property of the sampling distribution of the sample mean: the mean of the sampling distribution of the sample mean is equal to the population mean (μ).
State the formula for the mean of the sampling distribution of the sample mean: μ_x̄ = μ, where μ_x̄ is the mean of the sample means and μ is the population mean.
Substitute the given population mean (μ = 33.64 cm) into the formula to determine the mean of the sample means.
Conclude that the mean of all such sample means is equal to the population mean, which is 33.64 cm.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean refers to the distribution of means obtained from all possible samples of a specific size drawn from a population. According to the Central Limit Theorem, this distribution will be approximately normal if the sample size is sufficiently large, regardless of the population's distribution. The mean of this sampling distribution equals the population mean.
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Central Limit Theorem
The Central Limit Theorem (CLT) states that the distribution of the sample means will approach a normal distribution as the sample size increases, typically n ≥ 30 is considered sufficient. This theorem is crucial for making inferences about population parameters based on sample statistics, as it allows for the application of normal probability methods even when the original population distribution is not normal.
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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. In this case, since the population mean of arm circumferences is given as 33.64 cm, the mean of all sample means for samples of size 25 will also be 33.64 cm. This concept is fundamental in inferential statistics, as it underpins the reliability of sample estimates.
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