Question

Birth weights Researchers in Norway analyzed data on the birth weights of 400,000 newborns over a 6-year period. The distribution of birth weights is approximately Normal with a mean of 3668 grams and a standard deviation of 511 grams.

a. Sketch a graph that displays the distribution of birth weights for this population.

b. Sketch a possible graph of the distribution of birth weights for an SRS of size $5 . Calculate the range for this sample.

In this population, the range (Maximum - Minimum) of birth weights is 3417 grams. We technology to take 500 SRSs of size $n=5n=5$and calculate the range (Maximum Minimum) for each sample. The dotplot shows the results.

Short answer

(a)The middle $99.7\%$of a normal distribution lies with three standard deviation from the mean of the distribution.

(b)A possible sample of size 5 is then $2400,3000,3600,3700,4500\text{Range}=\max -\min =4500-2400=2100$

(c) The dots is the dot plot represent the sample range of simple random samples of size $n=5$. The dot at 2800 then represents a single simple random sample size $n=5$that has a sample ranges of 2800 grams.

(d) it is observed that all sample ranges are smaller than the population range of 3066 grams. This then implies that the centre of the sampling distribution of the sample ranges is not the population range and thus the sample range is not an unbiased estimator of the population range.

Step-by-step solution

Part (a) Step 1: Given Information

Given,

$$\begin{gathered} \mu =3668 \\ \sigma =511 \end{gathered}$$

Part (a) Step 2: Simplification

With one standard deviation from the distribution's mean, the centre 68 percent of a normal distribution lies.

The middle 95 percentile of a normal distribution is two standard deviations off the mean.

With three standard deviations from the distribution's mean, the middle 99.7% of a normal distribution is found.

Identifying values that are 1,2, and 3 standard deviations apart from the mean

$$\begin{gathered} \mu -3\sigma =3668-3\left(511\right)=2135 \\ \mu -2\sigma =3668-2\left(511\right)=2646 \\ \mu -\sigma =3668-511=3157 \\ \mu +\sigma =3668+511=4179 \\ \mu +2\sigma =3668+2\left(511\right)=4690 \\ \mu +3\sigma =3668+3\left(511\right)=5201 \end{gathered}$$

Part (b) Step 1: Given Information

Normal distribution

$$\begin{gathered} \mu =3668 \\ \sigma =511 \end{gathered}$$

Part (b) Step 2: Simplification

A basic random sample of size 5 has 5 data values that are broadly centred around a, with all data values falling between and. $\mu -3\sigma =3668-3\left(511\right)=2135$and $\mu +3\sigma =3668+3\left(511\right)=5201$.

The size of a probable sample of size 5 is then calculated.$2400,3000,3600,3700,4500$.

The Range $=\max -\min =4500-2400=2100$

Place a dot above the associating number on the number line for each given data value.

Part (c) Step 1: Given Information

Given , The graph is

Part (c) Step 2: Simplification

The dots on the dot plot show the sample range of simple random samples with a size of $n=5$. A single simple random sample size $n=5$with a sample range of 2800 grammes is represented by the dot at 2800.

Part (d) Step 1: Given Information

Given,

$$\begin{gathered} \mu =3668 \\ \sigma =511 \end{gathered}$$

Part (d) Step 2: Simplification

A normal distribution's middle 68 percent are one standard deviation from the distribution's mean. The middle 95 percentile of a normal distribution is two standard deviations off the mean. With three standard deviations from the distribution's mean, the middle 99.7% of a normal distribution is found.

$$\begin{gathered} \mu -3\sigma =3668-3\left(511\right)=2135 \\ \mu +3\sigma =3668+3\left(511\right)=5201 \end{gathered}$$

As a result, almost all data values fall between 5201 and 2135.

The difference between the maximum and minimum called the range.

$\text{Range}=\max -\min =5201-2135=3066$

This means that the population range is approximately 3066 grammes.

If the centre of the estimator's sample distribution equals the corresponding population parameter, the estimator is said to be unbiased.

All sample ranges are smaller than the population range of 3066 grammes, as seen in the dot plot. This suggests that the sample range's centre of sampling distribution is not the population range, and hence the sample range is not an unbiased estimator of the population range.