Question

Tall girls According to the National Center for Health Statistics, the distribution of height for $16$-year-old females is modeled well by a Normal density curve with mean $\mathrm{\mu }=64$inches and standard deviation $\mathrm{\sigma }=2.5$inches. Assume this claim is true for the three hundred $16$-year-old females at a large high school.

a. Make a graph of the population distribution.

b. Imagine one possible SRS of size $20$from this population. Sketch a dotplot of the distribution of sample data.

Short answer

Part(a) The graph

Part(b) The dot plot is

Step-by-step solution

Part(a) Step 1 : Given information

We need to draw graph for given data.

Part(a) Step 2 : Simplify

As given,

$$\begin{gathered} \mu =64 \\ \sigma =2.5 \end{gathered}$$

Now,

$68\%$lies in first standard deviation from mean.

$95\%$lies in second standard deviation from mean.

$99.7\%$lies in third standard deviation from mean.

Now, from the formula of standard deviation,

$$\begin{gathered} \mu -3\sigma =64-3(2.5)=56.5 \\ \mu -2\sigma =64-2(2.5)=59 \\ \mu -\sigma =64-(2.5)=61.5 \\ \mu +\sigma =64+(2.5)=66.5 \\ \mu +2\sigma =64+2(2.5)=69 \\ \mu +3\sigma =64+3(2.5)=71.5 \end{gathered}$$

The graph is

Part(b) Step 1 : Given information

We need to draw dotplot for the given data.

Part(b) Step 2 : Simplify

As given, Mean $\mathrm{\mu }=64$inches.

Standard deviation $\sigma =2.5$inches.

SRS size is $20$Now, the values that are centered are nearly $64$,

$$\begin{gathered} \mathrm{\mu }-3\mathrm{\sigma }=64-3(2.5)=56.5 \\ \mathrm{\mu }+3\mathrm{\sigma }=64+3(2.5)=71.5 \end{gathered}$$

The dotplot would be roughly a bell-shaped pattern.