Final answer:
The distribution of sample means will be approximately normal due to the Central Limit Theorem, even if the population is not normally distributed, provided the sample size is large enough (n ≥ 30). This applies to the question's scenario with n=78, allowing normal distribution calculations for sampling questions.
Step-by-step explanation:
The question relates to the concept of the sampling distribution of sample means, which is a fundamental concept in statistics, particularly when discussing the Central Limit Theorem (CLT). According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution, provided the sample size is large enough (typically n ≥ 30). Consequently, even if the population is skewed left, with the sample size of n=78 being larger than 30, the distribution of the sample means would be approximately normal with mean μ=50 and standard deviation σ/√n.
In the context of the questions provided:
- Although individual samples won't have the exact mean of the population (μ=50), their means will be clustered around 50 with an approximately normal distribution due to CLT.
- The standard deviation of the sampling distribution of sample means (σx) can be found by σ/√n, which would apply to a scenario where we repeatedly draw samples of size 100 from a population with a mean of 75 and standard deviation of 4.5.
- For finding probabilities related to sample means, such as the probability that a sample mean is between two values, the normal distribution can be used to calculate this, as shown in the example where n = 25, mean = 90, and standard deviation = 15.