Can we assume that the distribution of mean lead concentrations in blood collected from repeated samples of 52 police officers to be normal?

Question

BlackTom AI · Accepted Answer

The question asks about the distribution of the mean lead concentrations in blood from repeated samples of 52 officers. To unpack this, we should consider the central limit theorem and sampling distribution concepts.
Option 1: 'Yes, because we have more than 30 observations in a sample.' This is typically correct because the central limit theorem tells us that the sampling distribution of the sample mean becomes approximately normal as sample size grows, and n = 52 is larger than the common rule-of-thumb threshold of 30. This holds even if the underlying population distribution is not normal, assuming independence and random sampling.
Option 2: 'Yes, because the population distribution of lead concentration in blood is normal.' If the underlying population were perfectly normal, the sample mean distribution would be normal for any sample size, but the premise here is not given; we cannot rely on the population being normal. Therefore this justification is not required and may be misleading, since CLT provides the normality approximation even when the population is not normal.
Option 3: 'No, because the population distribution of lead concentration in blood is not normal.' This conflicts with the purpose of the CLT. A non-normal population does not prevent the sampling distribution of the mean from becoming approximately normal at moderate to large n. Thus this statement is incorrect as a general rule.
Option 4: 'Cannot be determined because the shape of the population distribution of lead concentration in blood is unknown.' While knowledge of the population shape is useful, the central limit theorem provides that with a reasonably large sample size, the sampling distribution of the mean is approximately normal regardless of the population shape (assuming independence). Therefore this option is not the best choice given n = 52.
In summary, the justification for Option 1 hinges on the central limit theorem: large samples yield a roughly normal sampling distribution of the mean, even if the population distribution is not normal, provided the data are independent and identically distributed. The other options either rely on an unstated population normality, deny the CLT implication, or ignore the established theorem that supports normality of the sample mean for large n.

Can we assume that the distribution of mean lead concentrations in blood collected from repeated samples of 52 police officers to be normal? 单项选择题

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