Understanding the Central Limit Theorem and Probabilities in Sampling Distributions

The Central Limit Theorem (CLT) is a fundamental concept in statistics that explains the distribution behavior of sample means obtained from a sufficiently large number of samples drawn from a population. This theorem is particularly important in inference, as it enables us to make probabilistic statements about the population mean based on the sample mean.

Case Study: A Population with Specific Characteristics

Consider a population with a mean μ 30 and a standard deviation σ 20. We are to calculate the probability that a random sample of size n 100 will have a sample mean between 28 and 34. This requires an understanding of the sampling distribution of the sample mean and how it is used to estimate probabilities.

Calculating the Mean and Standard Deviation of the Sample Mean

Step 1: Using the Central Limit Theorem, we recognize that the sampling distribution of the sample mean will be normally distributed since the sample size (n 100) is large.

Step 2: Compute the mean of the sample mean:

μbar{x} μ 30

Step 3: Calculate the standard deviation of the sample mean:

σbar{x} σ / √n 20 / 10 2

Standardizing the Sample Mean Values

Step 4: Convert the sample mean values to z-scores:

z1 (28 - 30) / 2 -1

z2 (34 - 30) / 2 2

Using the Standard Normal Distribution

Step 5: Find the probabilities corresponding to these z-scores using the standard normal distribution:

P(Z

P(Z

Calculating the Probability

Step 6: Calculate the probability that the sample mean is between 28 and 34:

P(28 ≤ bar{x} ≤ 34) P(Z

Final Answer: The probability that the sample mean is between 28 and 34 is approximately 0.8185 or 81.85%.

Specific Cases and Considerations

Let's consider two specific cases to further illustrate the concept:

Case 1: Population of 999,999 Values of 30.02 and 1 Value of -19970

In this scenario, the population is dominated by values close to 30.02, with only one extreme outlier. If a sample of 100 is taken, the chance of picking the outlier value (-19970) is very low at 0.0001. Therefore, the sample mean will almost certainly be in the interval 28 to 34, actually very close to 30.02 due to the prevalence of the 30.02 values.

Case 2: Population with a Gaussian Distribution

If the population has a Gaussian (normal) distribution, the probability that the sample mean will fall within the range 28 to 34 will remain approximately the same as calculated earlier using the CLT. This is due to the properties of the normal distribution and the properties of the sample mean as described by the CLT.

Concluding, the Central Limit Theorem is a statistical tool that enables us to make inferences about the population based on a sample, and its application can provide insights into the behavior of the sample mean and the probabilities associated with it. Understanding this theorem is crucial for advanced statistical analysis and real-world applications.