Final answer:
The question examines how sample size affects the shape of the sampling distribution and its approximation to normality. It involves calculating the sample mean, standard deviation, skewness, and comparing the distribution shapes for different sample sizes. Understanding the Central Limit Theorem is crucial for these analyses.
Step-by-step explanation:
The question pertains to the properties of sampling distributions, specifically how the sample size affects the approximation to a normal distribution. The application of the Central Limit Theorem (CLT) suggests that for sufficiently large sample sizes, the sampling distribution of the sample means will approach a normal distribution, regardless of the shape of the population distribution. However, the question mentions sample sizes of n=10 and n=5, which may not be large enough for this approximation to hold, especially if the population distribution is not normal or is highly skewed.
The determination of whether the distribution of sample means is approximately normal involves calculating the mean and standard deviation of the samples, analyzing the skewness, and comparing the impact of sample size on the distribution shape. For smaller samples, if the population distribution is not normal, the sample means may not be well approximated by a normal distribution.
Regarding the exercises mentioned, without specific numerical data, we cannot calculate the exact mean, standard deviation, or probabilities mentioned. However, we can state that:
- A smaller sample size generally leads to a greater standard deviation of the sampling distribution.
- A change in sample size can affect the shape of the sampling distribution due to the CLT, which states that larger samples will tend to produce a distribution that is more normal in shape.
- The probability of extreme sums or means can be calculated using the properties of the sampling distribution, which depends on the known mean and standard deviation of the population and the sample size.