Suppose you're in a class of 30 students. The instructor takes a simple random sample of five students and observes their ages. Imagine all of the different samples possible (there are 142,506 such samples possible!). Let denote the average age in your sample. The distribution of all values taken by in all possible samples of five students selected from the 30 students in your class is

Question

BlackTom AI · Accepted Answer

The sampling distribution of x̄ is the distribution of sample means computed from every possible sample of a given size drawn from the population.

Relevant knowledge points:
- A statistic (x̄) is a function of sample data; its sampling distribution is the set (or probability distribution) of all values that statistic would take over all possible samples of the specified size.
- The number of possible simple random samples of size 5 from 30 is "30 choose 5" = C(30,5) = 30·29·28·27·26 / (5·4·3·2·1) = 17,100,720 / 120 = 142,506 (matches the prompt).
- A parameter (like the population mean μ) is a fixed constant describing the population, not a distribution.
- The standard deviation is a single measure of spread, not the distribution of values.
- The phrase "probability that X is obtained" describes a single probability, not the distribution of sample means.

Step-by-step reasoning:
1. The instructor takes every possible simple random sample of size 5 from the class of 30. For each sample you can compute the sample mean x̄.
2. Collecting those means across all C(30,5)=142,506 samples produces a set (or distribution) of x̄ values — exactly the definition of a sampling distribution of the sample mean.
3. Therefore the distribution described in the problem is the sampling distribution of x̄.

Why option (a) is correct:
- Option (a) names exactly this concept: "the sampling distribution of x̄" — the distribution of all sample means across all possible samples.

Why the other options are incorrect:
- (b) "the parameter": a parameter is a single fixed population quantity (not a distribution of sample means).
- (c) "the probability that X is obtained": that would be a single probability value for a specific outcome, not the full distribution of sample means.
- (d) "the standard deviation of values": standard deviation is a single numeric summary of spread, not the full distribution.

Conclusion: The correct answer is a. the sampling distribution of x̄.

类似问题

What is a sampling distribution?

If you increase the sample size from 25 to 100 while keeping the population standard deviation constant, what happens to the standard deviation of the sampling distribution of the mean?

Which of the following statements is correct about a parameter or a statistic associated with repeated random samples of the same size from the same population?

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