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It is given that the mean birth weight of the babies under study is 3369 grams with a standard deviation of 581 grams.

The population, in this case, includes the babies whose birth weights are being measured.

In this case, the birth weight of the babies was measured. And also the birth weight varies from person to person. So the variable, in this case, is birth weight.

We know that the sample mean of a sample is equal to the population mean irrespective of the sample size.

The population mean in this case is given as $\mathrm{μ}=3369$grams.

So when the sample size includes $200$babies then the sample mean would be the same as the population mean.

Thus the mean of all possible sample mean weights of sample size $200$is $3369$grams.

We know that the sample standard deviation of a sample is equal to the standard deviation of the variable under consideration divided by the square root of the sample size.

It is given that the standard deviation of the weights is $\mathrm{σ}=581$grams.

So when the sample size is of $200$babies then the standard deviation is given as

$$\begin{gathered} {\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{200}} \\ {\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{581}{\sqrt{200}} \\ {\mathrm{σ}}_{\stackrel{¯}{x}}≈41.08 \end{gathered}$$

Thus the standard deviation of all possible sample mean weights of sample size $200$is$41.08$ grams.

We know that the sample mean of a sample is equal to the population mean irrespective of the sample size.

The population mean in this case is given as $\mathrm{μ}=3369$grams.

So when the sample size includes $400$babies then the sample mean would be the same as the population mean.

Thus the mean of all possible sample mean weights of sample size $400$is $3369$grams.

We know that the sample standard deviation of a sample is equal to the standard deviation of the variable under consideration divided by the square root of the sample size.

It is given that the standard deviation of the weights is $\mathrm{σ}=581$grams.

So when the sample size is of $400$babies then the standard deviation is given as

$$\begin{gathered} {\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{400}} \\ {\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{581}{\sqrt{400}} \\ {\mathrm{σ}}_{\stackrel{¯}{x}}=29.05 \end{gathered}$$

Thus the standard deviation of all possible sample mean weights of sample size $400$is $29.05$grams.