Answer :
Final answer:
The distribution of sample means for 200 adults is approximately normal due to the Central Limit Theorem, with a mean of 112 and a standard deviation of 1.414, making option (e) the correct choice.
Explanation:
The question asks about the distribution of the sample mean IQ of 200 randomly selected adults from a large population where the mean IQ is 112 with a standard deviation of 20. According to the Central Limit Theorem, when the sample size is large, the distribution of the sample means will be approximately normal (normal by the Central Limit Theorem), even if the source population itself is not perfectly normal.
The mean of the sampling distribution of the sample mean will be the same as the mean of the population, so the mean will be 112.
However, the standard deviation of the sampling distribution (often called the standard error) is equal to the standard deviation of the population divided by the square root of the sample size. So, the standard deviation for the sample mean for 200 adults would be 20 / √200, which is about 1.414. Therefore, the correct choice is (e) exactly Normal, mean 112, standard deviation 1.414.
Answer:
(c)approximately Normal, mean 112, standard deviation 1.414.
Step-by-step explanation:
To solve this problem, we have to understand the Central Limit Theorem
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 112, \sigma = 20, n = 200[/tex]
Using the Central Limit Theorem
The distribution of the sample mean IQ is approximately Normal.
With mean 112
With standard deviation [tex]s = \frac{20}{\sqrt{200}} = 1.414[/tex]
So the correct answer is:
(c)approximately Normal, mean 112, standard deviation 1.414.
