Final answer:
The statement is b) false because the mean and the standard deviation measure different things: one is about central tendency and the other is about data spread. A larger mean does not imply a larger standard deviation in the population.
Step-by-step explanation:
The statement 'For a normal population, a larger mean implies a larger standard deviation' is b) false.
The mean and the standard deviation are measures of different aspects of a distribution; the mean measures the central tendency, while the standard deviation measures the variability or dispersion of the data in a population.
A larger mean does not necessarily mean there will be a larger standard deviation.
The central limit theorem relates to the distribution of sample means, stating that as the sample size gets larger, the distribution of the sample means will approximate a normal distribution, regardless of the population distribution's shape.
This theorem also says that the standard deviation of the sampling distribution (the standard error) becomes smaller as the sample size increases.
The standard deviation is influenced by how spread out the data values are around the mean, not by the value of the mean itself.
Therefore, you can have two distributions with the same standard deviation but different means, or two distributions with the same mean and different standard deviations.
Facts like these highlight the independence of these two statistics.
The law of large numbers supports this, stating that as you take larger samples, the sample mean tends to get closer to the population mean, but this does not affect the population's standard deviation.