Question

weights of golden retriever dogs are normally distributed. samples of weights of golden retriever dogs, each of size ( n = 15 ), are randomly collected and the sample means are found. is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too small? explain.
choose the correct answer below.
a. yes; the sample size must be over 30 for the sample means to be normally distributed
b. no; the original population is normally distributed, so the sample means will be normally distributed for any sample size.
c. no; the samples are collected randomly, so the sample means will be normally distributed for any sample size.
d. no; as long as more than 30 samples are collected, the sample means will be normally distributed

Explanation:

The Central Limit Theorem states that if the original population is normally distributed, then the sampling distribution of the sample mean \(\bar{x}\) is also normally distributed for any sample size \(n\). In this case, the weights of golden retriever dogs (the original population) are normally distributed. So, regardless of the sample size \(n = 15\) (which is less than 30), the sample means will be normally distributed.

• Option A is incorrect because when the original population is normal, the \(n>30\) rule (which is for non - normal original populations) does not apply.
• Option C is incorrect. Random sampling is a good practice for sampling, but it is the normality of the original population (not just random sampling) that ensures the normality of the sample means for any sample size in this context.
• Option D is incorrect. The condition of the original population being normal makes the sample - mean distribution normal for any \(n\), not just when \(n>30\) (the \(n > 30\) is a rule - of - thumb for non - normal original populations).

Answer:

B. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.