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

Cem Hayes

Cem Hayes

Answered question

2021-03-09

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 the population distribution, no matter what the sample size is.
B. Sampling distributions of means are always nearly normal.
C. Sampling distributions of means get closer to normality as the sample size increases.
D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.

Answer & Explanation

un4t5o4v

un4t5o4v

Skilled2021-03-10Added 105 answers

Step 1
According to central limit theorem, sampling distribution of means approaches normal when the sample size increases.
Step 2
Therefore Sampling distributions of means get closer to normality as the sample size increases.
Option C is correct.
Jeffrey Jordon

Jeffrey Jordon

Expert2021-10-06Added 2605 answers

When the sample size is sufficiently large, obviously the sampling distributions get closer to normality. So option d is the correct answer.
Jazz Frenia

Jazz Frenia

Skilled2023-06-19Added 106 answers

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.
Andre BalkonE

Andre BalkonE

Skilled2023-06-19Added 110 answers

The correct statement about the sampling distribution of means is:
C. Sampling distributions of means get closer to normality as the sample size increases.
xleb123

xleb123

Skilled2023-06-19Added 181 answers

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.

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