Chapter 7, Problem R7.3RE

(a)

To determine

To construct: a graph that displays the distribution of birth weights for this population.

Explanation of Solution

Given:

  $\begin{array}{l}\mu =3668\\ \sigma =511\end{array}$

  

Calculation:

The middle $68\%$ of a normal distribution lies with one standard deviation from the mean of the distribution.

The middle $95\%$ of a normal distribution lies with two standard deviation from the mean of the distribution.

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

Graph:

  

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

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

  

(b)

To determine

To construct: a possible graph of the distribution of birth weights for an SRS of size 5 and calculate the range for this sample.

Explanation of Solution

Given:

Normal distribution

  $\begin{array}{l}\mu =3668\\ \sigma =511\end{array}$

  

Calculation:

A simple random sample of size 5 then contains 5 data values that are roughly centred about a, while all data values are between $\mu -3\sigma =3668-3\left(511\right)=2135$ and $\mu +3\sigma =3668+3\left(511\right)=5201$ .

A possible sample of size $5$ is then $2400,3000,3600,3700,4500$ .

  $\text{Range =max-min=4500-2400=2100}$

Graph:

Dot plot

  

For each given data value places a dot above the associating number on the number line.

(c)

To determine

To Explain: the value represents.

Explanation of Solution

Given:

  

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)

To determine

To Explain: the sample ranges an unbiased estimator of the population range and gives evidence from the graph to support the answer.

Explanation of Solution

Given:

  $\begin{array}{l}\mu =3668\\ \sigma =511\end{array}$

  

Calculation:

The middle $68\%$ of a normal distribution lies with one standard deviation from the mean of the distribution.

The middle $95\%$ of a normal distribution lies with two standard deviation from the mean of the distribution.

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

  $\mu -3\sigma =3668-3\left(511\right)=2135$

  $\mu +3\sigma =3668+3\left(511\right)=5201$ .

Thus nearly all data values are between 5201 and 2135.

The range is the difference between the maximum and minimum.

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

This then implies that the population range is about 3066 grams.

An estimator is called unbiased, if the centre of the sampling distribution of the estimator is equal to the corresponding population parameter.

In the dot plot, 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.