Question

Let x follow an approximate Normal model with centered at µ with a given standard deviation σ. If the xˉ follow an approximate Normal model with centered at μ with a given standard means (σxˉ) if the sample size was quadrupled (4n) ? The standard deviation of the sample means (σxˉ) would be 4 times greater The standard deviation of the sample means (σz) would be 2 times greater The standard deviation of the sample means (σx) would be halved. The standard deviation of the sample means (σxˉ) would be 1/4 its original value. The standard deviation of the sample means (σxˉ) would not change.

Answer 1

The standard deviation of the sample means, σX, is halved when the sample size is quadrupled according to the central limit theorem, which states σX = σ/√n.

Step-by-step explanation:

When determining how the standard deviation of the sample means, σX, changes as the sample size increases, we apply the central limit theorem.

According to the theorem, for a given population with a mean (μ) and standard deviation (σ), the standard deviation of the sample means (σX) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). This can be stated as σX = σ/√n.

Therefore, if the sample size is quadrupled from n to 4n, the new standard deviation of the sample means becomes σX = σ/√(4n)

= σ/(2√n).

Thus, the standard deviation of the sample means is halved when the sample size is quadrupled, not 4 times greater, 2 times greater, 1/4 of its original value, or unchanged.