Chapter 6.3, Problem 9E

New Residences The average number of moves a person makes in his or her lifetime is 12. If the standard deviation is 3.2, find the probability that the mean of a sample of 36 people is

a. Less than 10

b. Greater than 10

c. Between 11 and 12

To determine

To obtain: The probability that a random sample of 36 people is less than 10.

Answer to Problem 9E

The probability that a random sample of 36 people is less than 10 is 0.00009.

Explanation of Solution

Given info:

Average number of moves a person makes in his or her lifetime is $\mu =12$, standard deviation of number of moves a person makes in his or her lifetime is $\sigma =3.2$ and the sample size is $n=36$.

Calculations:

The random variable x represents number of moves of a person in his/her lifetime.

The notation $\overline{X}$ represents the mean number of moves of a person in his/her lifetime

The formula for finding the z score using central limit theorem is,

$z=\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Substitute 10 for $\overline{X}$, 12 for $\mu$, 3.2 for $\sigma$ and 36 for n

$\begin{array}{c}z=\frac{(10-12)}{\frac{3.2}{\sqrt{36}}}\\ =-3.75\end{array}$

The probability that the mean of the selected sample will be less than 10 moves represents the area to the left of ­3.75.

Software procedure:

Step-by-step procedure to obtain the probability using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as –3.75.
• Click OK.
• Output using the MINITAB software is given below:
• 

From MINITAB output, the probability that a random sample of 36 people is less than 10 is 0.00009.

To determine

To obtain: The probability that a random sample of 36 people is greater than 10.

Answer to Problem 9E

Probability will be 0.9999.

Explanation of Solution

Calculations:

The formula for finding the z score using central limit theorem is,

$z=\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Substitute 10 for $\overline{X}$, 12 for $\mu$, 3.2 for $\sigma$ and 36 for n

$\begin{array}{c}z=\frac{(10-12)}{\frac{3.2}{\sqrt{36}}}\\ =-3.75\end{array}$

The probability that a random sample of 36 people greater than 10 represents the area to the right of –3.75.

Software procedure:

Step-by-step procedure to obtain the probability using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Click the Shaded Area tab.
• Choose X value and Right Tail for the region of the curve to shade.
• Enter the X value as –3.75.
• Click OK.
• Output using the MINITAB software is given below:
• 
• From MINITAB output, the probability that a random sample of 36 people greater than 10 is 0.9999.

To determine

The probability that a random sample of 36 people between 11 and 12.

Answer to Problem 9E

The probability that a random sample of 36 people is between 11 and 12 is 0.4699.

Explanation of Solution

Calculations:

The formula for finding the z score using central limit theorem is,

$z=\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Substitute 11 and 12 for ${\overline{X}}_1$ and ${\overline{X}}_2$, 12 for $\mu$, 3.2 for $\sigma$ and 36 for n

$\begin{array}{l}{z}_1=\frac{(11-12)}{\frac{3.2}{\sqrt{36}}}\\ =-1.88\\ z_2=\frac{(12-12)}{\frac{3.2}{\sqrt{36}}}\\ =0\end{array}$

The probability that a random sample of 36 people is between 11 and 12 represents area between the value $-1.88$ and $0.$

Software procedure:

Step-by-step procedure to obtain the probability using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Click the Shaded Area tab.
• Choose X value and Middle for the region of the curve to shade.
• Enter the X value 1 as –1.88 and X value 2 as 0.
• Click OK.
• Output using the MINITAB software is given below:

From MINITAB output, the probability that a random sample of 36 people is between 11 and 12 is 0.4699.