Small Sample Weights of M&M plain candies are normally distributed. Twelve M&M plain candies are randomly selected and weighed, and then the mean of this sample is calculated. Is it correct to conclude that the resulting sample mean cannot be considered to be a value from a normally distributed population because the sample size of 12 is too small? Explain.

Verified step by step guidance

1. Understand the problem: The question is asking whether the sample mean of a small sample size (n = 12) can still be considered as coming from a normally distributed population. This involves understanding the relationship between sample size, normality, and the Central Limit Theorem.
2. Recall the Central Limit Theorem (CLT): The CLT states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution. However, if the population itself is already normally distributed, the sample mean will also be normally distributed, even for small sample sizes.
3. Identify the key information: The problem states that the weights of M&M plain candies are normally distributed. This means the population distribution is already normal, so the sample mean will also follow a normal distribution, regardless of the sample size (n = 12 in this case).
4. Address the concern about sample size: While a sample size of 12 is considered small, the normality of the population ensures that the sample mean is normally distributed. The sample size does not need to be large in this case because the population is already normal.
5. Conclude: It is correct to conclude that the sample mean can still be considered as coming from a normally distributed population, even with a small sample size of 12, because the population itself is normally distributed. The sample size does not invalidate this conclusion.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n ≥ 30). In this case, with a sample size of 12, while it may not be large enough for the theorem to fully apply, the underlying population is already normally distributed, which allows for valid conclusions about the sample mean.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this scenario, since the weights of M&M candies are stated to be normally distributed, any sample mean calculated from this population will also follow a normal distribution, even if the sample size is relatively small.

Sample Size and Statistical Inference

Sample size plays a crucial role in statistical inference, affecting the reliability of estimates and conclusions drawn from data. While larger samples generally provide more accurate estimates of population parameters, a sample size of 12 can still yield valid insights when drawn from a normally distributed population, allowing for reasonable conclusions about the sample mean despite its smaller size.