Final answer:
To obtain the sampling distribution of a continuous random variable such as an IQ score, multiple random samples of size n are taken from the population. The means of these samples create a distribution that approximates a normal distribution with a mean equal to the population mean and a variance based on sample size, due to the Central Limit Theorem.
Step-by-step explanation:
To understand the sampling distribution of a continuous random variable such as IQ scores, one needs to collect random samples of size n from the population. The means of these samples form the sampling distribution of the sample mean. According to the Central Limit Theorem, this distribution approaches a normal distribution as the sample size n increases, regardless of the shape of the original distribution.
Real-World Example: IQ Scores
Imagine a study on IQ scores across a certain population. A researcher might draw multiple random samples, each consisting of a specific number of individuals (n), and calculate the average IQ score for each sample. As more samples are collected and averages calculated, the distribution of these sample means will approximate a normal distribution centered around the population mean, with a standard deviation equal to the population standard deviation divided by the square root of n (standard error of the mean).
This illustrates how the population attribute, such as an IQ score, distributes itself among sample means when samples are taken repeatedly. The implication of this is critical for hypothesis testing and inferring population characteristics from sample statistics. With larger sample sizes, the sampling distribution becomes a powerful tool for making probabilistic predictions about population means.