Nationwide, the amount charged by doctors for performing a particular minor surgical procedure averages $1220 and varies with a standard deviation of $270. We randomly select 140 bills from the population of all bills charged for this surgery. Let represent the average amount charged for these 140 surgical procedures.The mean and standard deviation of the sample averages,, are

Question

BlackTom AI · Accepted Answer

For a sample mean, the expected value equals the population mean, so μ_X̄ = $1220.

1. Relevant knowledge points and concepts
- The sampling distribution of the sample mean X̄ has mean μ_X̄ = μ (the population mean).
- Its standard deviation (the standard error) is σ_X̄ = σ / √n, where σ is the population standard deviation and n is the sample size.
- With n = 140 (large), the Central Limit Theorem justifies treating the sampling distribution as approximately normal; independence is assumed by random sampling.

2. Complete reasoning process step by step
- Given population mean μ = $1220 and population standard deviation σ = $270.
- Sample size n = 140.
- Therefore μ_X̄ = μ = $1220.
- Compute the standard error: σ_X̄ = 270 / √140.

3. Calculations with intermediate steps
- √140 ≈ 11.832159569.
- σ_X̄ = 270 / 11.832159569 ≈ 22.82 (dollars).
- More explicitly: 270 ÷ 11.83216 ≈ 22.82, so the standard deviation of the sample mean is about $22.82.

4. Why the correct answer is correct
- Option (a) states mean = $1220 and standard deviation = $22.82, which exactly matches μ_X̄ = μ and σ_X̄ = σ/√n computed above. The units are dollars, and the numerical standard error result is obtained by dividing the individual-bill standard deviation by the square root of the sample size.

5. Why the other options are incorrect
- (b) and (d) list a mean of $122, which is off by a factor of 10 (incorrect population mean).
- (c) gives the standard deviation as $270, which is the individual-bill standard deviation, not the standard error of the sample mean; the sample mean’s standard deviation must be σ/√n, not σ.

6. Conclusion
- The mean and standard deviation of X̄ are μ_X̄ = $1220 and σ_X̄ ≈ $22.82, confirming option (a).

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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