Which of the following is true about the sampling distribution of means? A. Shape of the sampling distribution of means is always the same shape as th

Step 1: A. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is.
To evaluate this option, we need to consider the Central Limit Theorem (CLT). The CLT states that for a sufficiently large sample size, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution. However, for small sample sizes, the shape of the population distribution can influence the shape of the sampling distribution of the mean. Therefore, option A is not true in general.
Step 2: B. Sampling distributions of means are always nearly normal.
This option is a correct statement. As mentioned earlier, according to the Central Limit Theorem, for sufficiently large sample sizes, the sampling distribution of the mean is approximately normally distributed. This is true regardless of the shape of the population distribution. The approximation to normality improves as the sample size increases. Therefore, option B is true.
Step 3: C. Sampling distributions of means get closer to normality as the sample size increases.
This statement is also true. The Central Limit Theorem implies that as the sample size increases, the sampling distribution of the mean becomes increasingly close to a normal distribution. In other words, the distribution becomes more symmetrical and bell-shaped. Therefore, option C is true.
Step 4: D. Sampling distribution of the mean is always right-skewed since means cannot be smaller than 0.
This option is not true. The sampling distribution of the mean is not always right-skewed. It depends on the population distribution and the sample size. While it is true that the mean of a sample cannot be smaller than zero, this does not imply that the sampling distribution of the mean is always right-skewed. It can be symmetric or skewed in either direction, depending on the population distribution and sample size. Therefore, option D is not true.
In summary, the correct statements are options B and C. The sampling distributions of means are always nearly normal, and they get closer to normality as the sample size increases.

The correct statement about the sampling distribution of means is:
$C$. Sampling distributions of means get closer to normality as the sample size increases.

Answer:
B. Sampling distributions of means are always nearly normal.
C. Sampling distributions of means get closer to normality as the sample size increases.
Explanation:
Let's analyze each option and determine its correctness:
A. $Shape$ $of$ $the$ $sampling$ $distribution$ $of$ $means$ $is$ $always$ $the$ $same$ $shape$ $as$ $the$ $population$ $distribution$, $no$ $matter$ $what$ $the$ $sample$ $size$ $is.$
This statement is not true. The shape of the sampling distribution of means depends on the sample size. If the sample size is large enough (typically above 30) and the population distribution is not extremely skewed, the sampling distribution of means will be approximately normal.
B. $Sampling$ $distributions$ $of$ $means$ $are$ $always$ $nearly$ $normal.$
This statement is generally true. As mentioned above, if the sample size is sufficiently large and the population distribution is not highly skewed, the sampling distribution of means tends to follow a normal distribution.
C. $Sampling$ $distributions$ $of$ $means$ $get$ $closer$ $to$ $normality$ $as$ $the$ $sample$ $size$ $increases.$
This statement is true. As the sample size increases, the sampling distribution of means becomes more and more similar to a normal distribution. This property is known as the Central Limit Theorem.
D. $Sampling$ $distribution$ $of$ $the$ $mean$ $is$ $always$ $right$ $skewed$ $since$ $means$ $cannot$ $be$ $smaller$ $than$ $0.$
This statement is incorrect. The sampling distribution of the mean is not always right-skewed. It can be symmetrical or even left-skewed, depending on the population distribution and the sample size.