To make a statistical inference about a population based on a sample when the sample does not come from a normally distributed population, one key condition needs to be met. This specific condition ensures that the Central Limit Theorem can be applied, which allows us to make assumptions about the distribution of the sample mean regardless of the population distribution.
The Central Limit Theorem states that if the sample size is sufficiently large, the sampling distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution. The general rule of thumb for this is that the sample size [tex]\( n \)[/tex] should be at least 30.
Here are the options provided:
1. [tex]\(\mu \geq 30\)[/tex]
2. [tex]\(\rightarrow 20\)[/tex]
3. [tex]\(n \geq 30\)[/tex]
4. [tex]\(N \geq 30\)[/tex]
Let's evaluate each option:
1. [tex]\(\mu \geq 30\)[/tex]: The symbol [tex]\(\mu\)[/tex] represents the population mean. This condition is unrelated to the requirement for the sample size.
2. [tex]\(\rightarrow 20\)[/tex]: This is not a standard notation or a meaningful condition in this context.
3. [tex]\(n \geq 30\)[/tex]: This condition states that the sample size [tex]\( n \)[/tex] must be at least 30. This aligns with the Central Limit Theorem, ensuring that the sampling distribution of the sample mean is approximately normal.
4. [tex]\(N \geq 30\)[/tex]: [tex]\( N \)[/tex] typically denotes the population size or another parameter, but in this context, the sample size is the critical factor, not the population size.
Based on the explanation, the correct condition that must be met is [tex]\( n \geq 30 \)[/tex]. Therefore, the correct answer is:
[tex]\[ n \geq 30 \][/tex]
This answer ensures that you can make a valid statistical inference about a population based on a sample that does not come from a normally distributed population.
