Question

For which of the following situations would the central limit theorem not imply that the sample distribution for x¯x¯ is approximately Normal?A. a population is not Normal, and we use samples of size ????=50n=50 .B. a population is not Normal, and we use samples of size ????=6n=6 .C. a population is Normal, and we use samples of size ????=50n=50 .D. a population is Normal, and we use samples of size ????=6n=6 .

Answer 1

Answer:

B. a population is not Normal, and we use samples of size n=6 .

Not satisfy the conditions since the sample size is not large enough

Step-by-step explanation:

Previous concepts

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

From the central limit theorem we know that the distribution for the sample mean $\bar X$ is given by:

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

Solution to the problem

The assumptions to use the central limit theorem are:

a) We need that the data would be sampled randomly.

b) the Samples should be independent of each other.  

c) The sample size should be sufficiently large (n>30).

A. a population is not Normal, and we use samples of size n=50 .

Satisfy the conditions since the no matter if the distribution of X is not normal, we have that the sample size is enough large

B. a population is not Normal, and we use samples of size n=6 .

Not satisfy the conditions since the sample size is not large enough

C. a population is Normal, and we use samples of size n=50 .

Satisfy alll the conditions, sample size large enough

D. a population is Normal, and we use samples of size n=6 .

The sample size is not large enough but on this case the distribution for X is normal so then would be apply apply the CLT.