Rolling a fair ten$-$sided die produces a uniformly distributed set of numbers between $1$ and $10$ with a mean of $5\mathrm{.}5$ and a standard deviation of $2\mathrm{.}872.$ Assume that n ten$-$sided dice are rolled many times and the mean of the n outcomes is computed each time.

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Part $1$

a$.$ Find the mean and standard deviation of the resulting distribution of sample means for n$=121.$

The mean of the resulting distribution of sample means is enter your response here.

$($Round to the nearest tenth as$$ needed.$)$

Part $2$

The standard deviation of the distribution of sample means is enter your response here.

$($Round to the nearest thousandth as$$ needed.$)$

Part $3$

b$.$ Find the mean and standard deviation of the resulting distribution of sample means for n$=36.$

The mean of the resulting distribution of sample means is enter your response here.

$($Round to the nearest tenth as$$ needed.$)$

Part $4$

The standard deviation of the distribution of sample means is enter your response here.

$($Round to the nearest thousandth as$$ needed.$)$

Part $5$

c$.$ Why is the standard deviation in part a different from the standard deviation in part$$ b$?$ Choose the correct answer below.

A$.$

With smaller sample sizes$($as in part$$ b$),$ the means tend to be closer$$ together, so they have less$$ variation, which results in a smaller standard deviation.

B$.$

With larger sample sizes$($as in part$$ a$),$ the means tend to be closer$$ together, so they have less$$ variation, which results in a smaller standard deviation.

C$.$

With larger sample sizes$($as in part$$ a$),$ the means tend to be further$$ apart, so they have more$$ variation, which results in a bigger standard deviation.

D$.$

With smaller sample sizes$($as in part$$ b$),$ the means tend to be further$$ apart, so they have more$$ variation, which results in a smaller standard deviation.