There is variation in the amount of water put in 24-ounce bottles when filled on the assembly line. Amounts are skewed left with mean,
𝜇
=
24.1
oz, and standard deviation
𝜎
=
0.4
oz. If repeated random samples of size 36 bottles are pulled off the assembly line, what is the expected mean of the sampling distribution of
𝑋
¯
?
Round your answer to the nearest hundredths place.

Question

There is variation in the amount of water put in 24-ounce bottles when filled on the assembly line. Amounts are skewed left with mean,  
𝜇
=
24.1
 oz, and standard deviation 
𝜎
=
0.4
 oz. If repeated random samples of size 36 bottles are pulled off the assembly line, what is the expected mean of the sampling distribution of 
𝑋
¯
?
 Round your answer to the nearest hundredths place.

BlackTom AI · Accepted Answer

The sampling distribution of the sample mean has mean equal to the population mean, μ.

1. Relevant knowledge points and concepts
- Sampling distribution of the sample mean: its expected value (mean) equals the population mean μ, regardless of sample size, provided samples are random.
- Central Limit Theorem (CLT): for sufficiently large n, the sampling distribution is approximately normal even if the original population is skewed. Here n = 36, which is commonly considered large enough.
- Standard error (SE) is σ/√n; this describes spread, not the expected mean.

2. Step-by-step reasoning
- We are asked for the expected mean of the sampling distribution of X̄ (the sample mean) for samples of size n = 36.
- By definition, E[X̄] = μ (the population mean). The shape or skewness of individual observations does not change this equality.

3. Calculations with intermediate steps
- Given population mean μ = 24.1 oz.
- Therefore E[X̄] = μ = 24.1 oz.
- If showing the standard error for completeness: σ = 0.4 oz, n = 36, √n = √36 = 6, so SE = σ/√n = 0.4/6 ≈ 0.0666667 oz. (This is not the expected mean but the spread of X̄.)
- Rounding the mean to the nearest hundredths place: 24.1 → 24.10.

4. Why the correct answer is correct
- The equality E[X̄] = μ is a fundamental property of the sample mean; thus the expected value of the sampling distribution equals the given population mean 24.1 oz. The CLT supports using normal approximation for shape but is not needed to determine the mean.

5. Other possible mistakes (why alternatives would be incorrect)
- Confusing the expected mean with the standard error (e.g., subtracting or adding SE to μ) would yield incorrect values like 24.03 or 24.17; these are not the expected mean.
- Using some adjusted biased estimator is unnecessary; the sample mean is unbiased, so the correct expectation remains μ.

6. Conclusion
- The expected mean of the sampling distribution of X̄ is 24.10 oz, which matches the provided answer 24.10.

类似问题

Suppose that Quackers now runs 200 people in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

Suppose that Quackers only runs 1 person in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

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?

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