M&M’s The candy company claims that 10% of the M&M’s it produces are green. Suppose the
candies are packed at random into small bags containing about 25 M&M’s per bag. A class of
elementary students learning about percentages opens several bags, counts the various colors, and
calculates the proportion that are green. Simultaneously, a second class performs the same
investigation but their bags contain about 50 M&M’s per bag.
Which class (large bags or small bags) can use the CLT and why?
I don't understand this can you even use the cental limit theorem for this? and how?
Yes, the Central Limit Theorem (CLT) can be used in this scenario, but it is more applicable to the class that opens the large bags. Let me explain why:
The Central Limit Theorem states that, under certain conditions, the distribution of sample means tends to be approximately normally distributed, regardless of the shape of the population distribution. One of the conditions required for the CLT to hold is that the samples are independent and identically distributed (i.i.d.).
In this case, the class that opens the large bags (containing about 50 M&M's per bag) is more likely to have i.i.d. samples compared to the class that opens the small bags (containing about 25 M&M's per bag). This is because the larger sample size reduces the impact of any individual bag's proportion of green M&M's, making it more likely that the samples are roughly independent and have the same probability distribution.
Furthermore, the sample size of the large bags is generally considered large enough (around 50) to satisfy the minimum sample size requirement for the CLT. Small sample sizes, on the other hand, may not satisfy this condition.
Therefore, the class that opens the large bags can use the Central Limit Theorem to make inferences about the proportion of green M&M's in the population, as the distribution of sample means for their larger sample size will be more likely to approximate a normal distribution.