Final answer:
The conditions for the Central Limit Theorem (CLT) are met because the sample size is large enough (50 words), the samples are likely independent, and there is a finite variance in word lengths from actual speeches. Assuming the standard deviation is the same for both justices' speeches, the CLT can be applied to compare average word lengths.
Step-by-step explanation:
The question asks whether the conditions for the Central Limit Theorem (CLT) are met for comparing word lengths in speeches by Justice Sonia Sotomayor and another Supreme Court Justice, given a random sample of 50 words and an average word length of 4.1. The standard deviation is assumed to be the same for both justices' speeches.
The CLT conditions are:
- A sufficiently large sample size
- Independent samples
- Sampling from a population with a finite level of variance
For the CLT to apply, the sample size should be large enough (usually n ≥ 30). In this case, with a sample size of 50 words, the first condition is met. Independence means that the selection of one item doesn't affect the selection of another. Assuming speeches are long and words are well-varied, this criterion is likely met as well. Lastly, since we're comparing word lengths that come from actual speeches, there's a finite level of variance, thus meeting the third condition.
Hence, if the standard deviation is known and assumed to be the same, all conditions for the Central Limit Theorem appear to be satisfied, and it can be applied for making inferences about the average word lengths in the speeches of the justices.