Answer :
Final answer:
Yes, the 68-95-99.7 rule (Empirical Rule) applies to the distribution of sample means with a standard deviation of 1.0, as the Central Limit Theorem suggests that the distribution of sample means is approximately normal for large samples such as 100.
Explanation:
The question is whether the 68-95-99.7 rule, also known as the Empirical Rule, applies to a distribution of sample means with a standard deviation of 1.0. The Empirical Rule states that for a bell-shaped and symmetric distribution, approximately 68% of data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
According to the Central Limit Theorem, the distribution of sample means will be approximately normal (bell-shaped) if the sample size is large enough, regardless of the population distribution from which samples are drawn. For samples of size 100, the theorem would suggest that the distribution of sample means is normal. Therefore the Empirical Rule would apply, and it's true that approximately [tex]68%[/tex] of sample means would fall within one standard deviation (1.0 in this case) of the overall mean of the sample means.
