Q. 6.95 - Elementary Statistics

Question

A variable is normally distributed with mean $10$ and standard deviation $3$.

Part (a): Determine and interpret the quartiles of the variable.

Part (b): Obtain and interpret the seventh percentile.

Part (c): Find the value that $35\%$of all possible values of the variable exceed.

Part (d): Find the two values that divide the area under the corresponding normal curve into a middle area of $0.99$ and two outside areas of $0.005$. Interpret your answer.

Step-by-Step Solution

Answer

Part (a): The first percentile is $7.98$. Therefore,$25\%$of the x-values below $7.98$ and $75\%$ values above $7.98$.

The second percentile is $10$. Therefore, $50\%$of the x-values below $10$ and $50\%$values above $10$.

The third percentile is $12.02$. Therefore, $75\%$ of the x-values below $12.02$ and $25\%$ values above $12.02$.

Part (b): $70\%$of the x-values below $11.57$ and $30\%$ values above $11.57$.

Part (c): The value that $35\%$ of all possible values of the variable exceed is $11.16$.

Part (d): $99\%$of all observations are between $2.27$ and $17.73$.

Part (a) Step 1. Given information.

The given mean is $10$and standard deviation is $3$.

Part (a) Step 2. Determine the first quartile of the variable.

Sketch a normal curve using $$\begin{gathered} \mu =10, \\ \sigma =3 \end{gathered}$$

Shade the region corresponding to first quartile or $25$th percentile.

We need to find the z-score corresponding to $P_{25}$ is the one having an area of $0.25$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $-0.67$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(-0.67\right) \\ =7.98 \end{gathered}$$

On interpreting, we can say, $25\%$ of the x-values below $7.98$ and $75\%$ values above $7.98$.

Part (a) Step 3. Determine the second quartile of the variable.

Shade the region corresponding to second quartile or $50$th percentile.

We need to find the z-score corresponding to $P_{50}$ is the one having an area of $0.5$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $0$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(0\right) \\ =10 \end{gathered}$$

On interpreting, we can say, $50\%$ of the x-values below $10$ and $50\%$ values above $10$.

Part (a) Step 4. Determine the third quartile of the variable.

Shade the region corresponding to second quartile or $75$th percentile.

We need to find the z-score corresponding to $P_{75}$ is the one having an area of $0.75$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $0.67$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(0.67\right) \\ =12.02 \end{gathered}$$

On interpreting, we can say, $75\%$of the x-values below $12.02$ and $25\%$ values above $12.02$.

Part (b) Step 1. Determine the seventh percentile.

Shade the region corresponding seventh percentile.

We need to find the z-score corresponding to $P_{70}$ is the one having an area of $0.52$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $0.52$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(0.52\right) \\ =11.57 \end{gathered}$$

On interpreting, we can say, $70\%$ of the x-values below $11.57$ and $30\%$ values above $11.57$.

Part (c) Step 1. Determine the value that 35 % of all possible values of the variable exceed.

First, we find the z-score corresponding to $P_{65}$is the one having an area of $0.39$to its left under the standard normal curve. From the normal distribution tables, the z-score is $0.39$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(0.39\right) \\ =11.16 \end{gathered}$$

Part (d) Step 1. Find the two values.

Shade the region corresponding to $0.5$th percentile.

We need to find the z-score corresponding to $P_{0.5}$ is the one having an area of $0.005$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $-2.576$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(-2.576\right) \\ =2.27 \end{gathered}$$

Part (d) Step 2. Determine the 99 . 5 th percentile of the variable.

Shade the region corresponding to second quartile or $99.5$th percentile.

We need to find the z-score corresponding to $P_{99.5}$ is the one having an area of $0.995$ to its left under the standard normal curve. From the normal distribution tables, the z-score is $2.576$.

On finding the value of x,

$$\begin{gathered} x=\mu +\sigma z \\ =10+3\left(2.576\right) \\ =17.73 \end{gathered}$$

On interpreting, we can say, $99\%$of all observations are between $2.27$ and $17.73$.