Suppose a population has probability distribution with a mean
𝜇
and a standard deviation
𝜎
. If a random sample of 100 measurements are taken from this population, you would expect very few of them to fall within 3 standard deviations of the mean.

c07.p066.q001

Question

Suppose a population  has  probability  distribution  with a mean 
𝜇
 and a standard deviation 
𝜎
. If a random sample of 100 measurements are taken from this population, you would expect very few of them to fall within 3 standard deviations of the mean.

c07.p066.q001

BlackTom AI · Accepted Answer

Let’s unpack the statement by recalling a fundamental rule of statistics known as the Empirical Rule or 68-95-99.7 rule. It tells us how data in a normal distribution typically spread around the mean in terms of standard deviations.
Option 1: True. This option asserts that very few observations would fall within 3 standard deviations of the mean. Under the Empirical Rule, about 99.7% of observations from a normal distribution lie within ±3σ of the mean, so this claim is incorrect. Saying “very few” fall within 3σ contradicts the well-established guideline that nearly all values are contained there.
Option 2: False. This is the correct interpretation for a population with a roughly normal distribution, or any distribution where the rule is a good approximation. Since 99.7% fall inside ±3σ, not “very few,” the statement mischaracterizes the typical spread of data.
Additional nuance: even if the underlying distribution isn’t perfectly normal, for a large sample like n = 100, the majority of sample measurements are still expected to lie within three standard deviations from the mean unless the distribution has extremely heavy tails. Therefore, the claim that very few fall within 3σ is generally not supported by standard probabilistic intuition.
In summary, option 1 is incorrect because empirical data show a large proportion within ±3σ, and option 2 is correct because it aligns with that understanding, indicating that the original statement (that very few fall within 3σ) is false.

Suppose a population has probability distribution with a mean 𝜇 and a standard deviation 𝜎 . If a random sample of 100 measurements are taken from this population, you would expect very few of them to fall within 3 standard deviations of the mean. c07.p066.q001判断题

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