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 will the shape of the of the sampling distribution of
𝑋
¯
be?

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 will the shape of the of the sampling distribution of 
𝑋
¯
 be?

BlackTom AI · Accepted Answer

By the Central Limit Theorem, the sampling distribution of the sample mean tends toward a normal distribution as sample size increases, regardless of the shape of the population distribution.

1. Relevant knowledge points and concepts
- Central Limit Theorem (CLT): For sufficiently large sample sizes, the distribution of sample means X̄ is approximately normal, with mean µX̄ = µ and standard error σX̄ = σ/√n.
- Standard error: quantifies variability of sample means around the population mean.
- Rule of thumb: n ≥ 30 is often considered large enough for the CLT to apply even when the population is skewed.

2. Step-by-step reasoning
- Population mean µ = 24.1 oz and population standard deviation σ = 0.4 oz.
- Sample size n = 36, which meets the typical "large enough" threshold for the CLT.
- Although the individual bottle amounts are skewed left, the CLT implies the distribution of the sample mean will be approximately normal for n = 36.

3. Calculations (all intermediate steps)
- Compute the standard error:
  σX̄ = σ / √n
  √n = √36 = 6
  σX̄ = 0.4 / 6 = 0.066666... ≈ 0.0667 oz
- The sampling distribution is centered at µX̄ = 24.1 oz with standard error ≈ 0.0667 oz.

4. Why the correct answer is correct
- Because n = 36 is sufficiently large, the CLT ensures the sampling distribution of X̄ is approximately normal, despite the population being skewed left. The mean of the sampling distribution equals the population mean and the spread is reduced to the standard error computed above.

5. Why the other options are incorrect
- "Skewed left": would describe the original population, but not the distribution of sample means for large n.
- "Bimodal": would require the underlying population to be bimodal and n small; CLT for n = 36 rules this out.
- "Skewed right": incorrect sign and same reasoning as "skewed left" — the sampling distribution will not retain the population skew for large samples.

6. Conclusion
The sampling distribution of X̄ will be approximately normal (correct answer: Approximately normal).

Similar Questions

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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