If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means x taken from random samples of a given size is normal?

Yes. Sample size need to be at least 30.

Step-by-step explanation:

The central limit theorem states that the sampling distribution of a sample mean is approximately normal if the sample size is large enough(≥30), even if the population distribution is not normal.

The central limit theorem also states that the sampling distribution will have the following properties

1. The mean of the sampling distribution will be equal to the mean of population distribution.

2. The standard deviation of the sampling distribution will be equal to the variance of the population distribution divided by the sample size.

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step-by-step explanation: