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.
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7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
1:21 minutesProblem 5.4.1
Textbook Question
In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.
Mu = 150, sigma =25, n = 49
Verified step by step guidance1
Step 1: Recall the formula for the mean of the sampling distribution of sample means. The mean of the sampling distribution (denoted as μₓ̄) is equal to the population mean (μ). Therefore, μₓ̄ = μ.
Step 2: Substitute the given value of the population mean (μ = 150) into the formula. This means the mean of the sampling distribution is μₓ̄ = 150.
Step 3: Recall the formula for the standard deviation of the sampling distribution of sample means. The standard deviation of the sampling distribution (denoted as σₓ̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). The formula is: σₓ̄ = σ / √n.
Step 4: Substitute the given values into the formula for the standard deviation. Use σ = 25 and n = 49. This gives: σₓ̄ = 25 / √49.
Step 5: Simplify the expression for the standard deviation. Calculate the square root of 49 (which is 7) and divide 25 by 7 to find the value of σₓ̄. This will give you the standard deviation of the sampling distribution.
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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 of the sample mean is the probability distribution of all possible sample means from a population. It describes how the means of different samples will vary and is crucial for understanding the behavior of sample statistics. The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution will approach a normal distribution as the sample size increases.
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Mean of the Sampling Distribution
The mean of the sampling distribution of the sample means, also known as the expected value, is equal to the population mean (mu). This means that if you take many samples and calculate their means, the average of those means will converge to the population mean. In this case, with mu = 150, the mean of the sampling distribution will also be 150.
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Standard Deviation of the Sampling Distribution (Standard Error)
The standard deviation of the sampling distribution, often referred to as the standard error, measures the variability of sample means around the population mean. It is calculated by dividing the population standard deviation (sigma) by the square root of the sample size (n). For this problem, with sigma = 25 and n = 49, the standard error can be computed to understand how much sample means will typically deviate from the population mean.
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