Question

In which of the following scenarios would the distribution of the sample mean x-bar be normally distributed? Check all that apply. We take repeated random samples of size 10 from a population of unknown shape. We take repeated random samples of size 15 from a population that is normally distributed. We take repeated random samples of size 50 from a population of unknown shape. We take repeated random samples of size 25 from a population that of unknown shape.

Answer 1

We take repeated random samples of size 10 from a population of unknown shape. INCORRECT, the sample isn't large enough to approximate the sample mean distribution to normal.

We take repeated random samples of size 15 from a population that is normally distributed. IS CORRECT, the sample mean will have a normal distribution.

We take repeated random samples of size 50 from a population of unknown shape. IS CORRECT, even though the variable has unknown distribution, the sample taken is large enough to approximate the sample mean distribution to normal.

We take repeated random samples of size 25 from a population that of unknown shape. INCORRECT, the sample isn't large enough to approximate the sample mean distribution to normal.

Explanation:

Hello!

Remember that the sample mean will always have the same distribution as the study variable. So if you take a sample and measure a variable of interest (X), and this variable has a normal distribution, the calculated sample mean will also have a normal distribution:

• X~N(μ;σ²) ⇒ X[bar]~N(μ;σ²/n)

We take repeated random samples of size 15 from a population that is normally distributed. IS CORRECT, the sample mean will have a normal distribution.

Next, to choose the correct answer you need to apply the Central Limit Theorem.

This theorem states that to any population of unknown distribution or known not normal distribution if the sample is large enough (most commonly accepted as "large enough samples" are those equal or greater than 30) the distribution of the sample mean tends to a normal distribution:

• X~?; n≥30 ⇒ X[bar]≈N(μ;σ²/n)

So, for the three premises below,

We take repeated random samples of size 10 from a population of unknown shape. INCORRECT, the sample isn't large enough to approximate the sample mean distribution to normal.

We take repeated random samples of size 50 from a population of unknown shape. IS CORRECT, even though the variable has unknown distribution, the sample taken is large enough to approximate the sample mean distribution to normal.

We take repeated random samples of size 25 from a population that of unknown shape. INCORRECT, the sample isn't large enough to approximate the sample mean distribution to normal.

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