Q. 7.47 - Elementary Statistics

Question

Baby Weight. The paper "Are Babies Normal?" by T. Clemons and M. Pagano (The American Statistician, Vol. 53, No, 4. pp. 298-302) focused on birth weights of babies. According to the article, the mean birth weight is 3369 grams (7 pounds, 6.5 ounces) with a standard deviation of 581 grams.
a. Identify the population and variable.
b. For samples of size 200, find the mean and standard deviation of all possible sample mean weights.
c. Repeat part (b) for samples of size 400.

Step-by-Step Solution

Answer

Part a. The population includes the babies and the variable includes the birth weight of the babies.

Part b. The mean and standard deviation of all possible sample mean weights for samples of size 200 are 3369 grams and 41.08 grams.

Part c. The mean and standard deviation of all possible sample mean weights for samples of size 400 are 3369 grams and 29.05 grams.

Part (a) Step 1. Given Information

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

Part (a) Step 2. Identify the population and the variable

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. 

Part (b) Step 1. Find the mean for the sample

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 $\mu =3369$ 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.

Part (b) Step 2. Find the standard deviation

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 $\sigma =581$ grams.

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

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{200}} \\ {\sigma }_{\overline{x}}=\frac{581}{\sqrt{200}} \\ {\sigma }_{\overline{x}}\approx 41.08 \end{gathered}$$

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

Part (c) Step 1. Find the mean for the sample

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 $\mu =3369$ 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.

Part (c) Step 2. Find the standard deviation

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 $\sigma =581$ grams.

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

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{400}} \\ {\sigma }_{\overline{x}}=\frac{581}{\sqrt{400}} \\ {\sigma }_{\overline{x}}=29.05 \end{gathered}$$

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