Suppose a population has an unknown probability distribution with a mean
𝜇
and a standard deviation
𝜎
.

TRUE/FALSE

An advantage of using a sample mean
𝑋
¯
based on many many observations over a single observation of X is that the probability distribution of
𝑋
¯
is known ( approximately ) even when the distribution of X is not.

c07.p066.q005

Question

Suppose a population  has an unknown  probability  distribution  with a mean 
𝜇
 and a standard deviation 
𝜎
.

TRUE/FALSE

An advantage of using a sample mean 
𝑋
¯
 based on many many observations over a single observation of  X is that the probability distribution of 
𝑋
¯
 is known ( approximately ) even when the distribution of X is not.

c07.p066.q005

BlackTom AI · Accepted Answer

The statement concerns the distribution of the sample mean X̄ when drawn from a population with unknown distribution but known mean μ and standard deviation σ. 
Option 1 (True): This is correct in the sense that as the sample size n grows, the sampling distribution of X̄ becomes approximately normal with mean μ and standard deviation σ/√n, due to the Central Limit Theorem. Thus, even if the original distribution of X is not known to be normal, the distribution of X̄ is (approximately) known for sufficiently large n, which is an advantage of using the average of many observations instead of a single observation. This justification relies on the CLT and the law of large numbers for stability of the mean estimates. 
Option 2 (False): This would be incorrect because it contradicts the central principle that X̄ has a known (approximately normal) distribution only under the CLT for large n; it also depends on sample size. If n is small or the underlying distribution is highly nonnormal, the distribution of X̄ may not be well approximated by a known form, so claiming it is always known for any sample size would be inaccurate. 
In summary, the stated advantage hinges on the CLT: with many observations, the sampling distribution of the mean becomes approximately normal with mean μ and standard error σ/√n, enabling inference even when the parent distribution is unknown. The counterpoint emphasizes that this approximation improves with larger n and under certain regularity conditions; it is not guaranteed for every possible X distribution and every n.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture #20 ( 92 - 93, 99 - 101 )

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