Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

Verified step by step guidance

1. Step 1: Recall the formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error of the mean). It is given by: $\sigma$_mean = σ / n, where $\sigma$ is the population standard deviation and $n$ is the sample size.
2. Step 2: Identify the values needed for the formula. From the problem, the sample size $n$ is 36. You will also need the population standard deviation $\sigma$, which should be provided in Exercise 5 or the dataset.
3. Step 3: Use technology (such as a calculator, spreadsheet software, or statistical software) to compute the standard deviation of the sample means. Plug the values of $\sigma$ and $n$ into the formula: $\sigma$_mean = σ / n.
4. Step 4: Compare the computed standard deviation of the sample means to the standard deviation of the ages found in Exercise 5. Note that the standard deviation of the sample means should be smaller than the population standard deviation, as predicted by the Central Limit Theorem.
5. Step 5: Reflect on the Central Limit Theorem, which states that the standard deviation of the sampling distribution of the sample mean decreases as the sample size increases. Verify that the result aligns with this prediction, as the sample size of 36 is relatively large.

Key Concepts

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. It is crucial for understanding the variability of sample means in relation to the population from which they are drawn.

Central Limit Theorem (CLT)

The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is fundamental in statistics because it allows for the use of normal probability techniques to make inferences about population parameters based on sample statistics, especially when dealing with large samples.

Sample Means

Sample means are the averages calculated from subsets of a population. When multiple samples are taken, each will yield a mean, and analyzing these sample means can provide insights into the overall population mean. The standard deviation of these sample means, known as the standard error, is essential for understanding how much the sample means are expected to vary from the true population mean.