Question

A random sample is selected from a population with mean μ = 102 and standard deviation σ = 10. For which of the sample sizes would it be reasonable to think that the x sampling distribution is approximately normal in shape? (Select all that apply.) a. n = 14 b. n = 19 c. n = 45 d. n = 55 e. n = 110 f. n = 440

Answer 1

Answer:

Since we can't assume that the distribution of X is the normal then we need to apply the central limit theorem in order to approximate the $\bar X$ with a normal distribution. And we need to check if n>30 since we need a sample size large as possible to assume this.

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

Based on this rule we can conclude:

a. n = 14 b. n = 19 c. n = 45 d. n = 55 e. n = 110 f. n = 440

Only for c. n = 45 d. n = 55 e. n = 110 f. n = 440 we can ensure that we can apply the normal approximation for the sample mean

for n=14 or n =19 since the sample size is <30 we don't have enough evidence to conclude that the sample mean is normally distributed

Step-by-step explanation:

For this case we know that for a random variable X we have the following parameters given:

$\mu = 102, \sigma =10$

Since we can't assume that the distribution of X is the normal then we need to apply the central limit theorem in order to approximate the $\bar X$ with a normal distribution. And we need to check if n>30 since we need a sample size large as possible to assume this.

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

Based on this rule we can conclude:

a. n = 14 b. n = 19 c. n = 45 d. n = 55 e. n = 110 f. n = 440

Only for c. n = 45 d. n = 55 e. n = 110 f. n = 440 we can ensure that we can apply the normal approximation for the sample mean

for n=14 or n =19 since the sample size is <30 we don't have enough evidence to conclude that the sample mean is normally distributed