Chapter 11.5, Problem 8CYU

(a)

To determine

To find: The probability for $P\left(X<2\right)$ , and then sketch the area under curve.

Answer to Problem 8CYU

The required probability is $0.3445$ , and graph is shown in figure (4).

Explanation of Solution

Given:

The number of sample is $925$ .

The mean is $1.8$ .

The standard deviation is $0.5$ .

Calculation:

Calculate the corresponding $z$ value for $X=2$ as follows.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{2-1.8}{0.5}\\ =0.4\end{array}$

Calculate the required probability by using TI-83 calculator, and sketch the area between $z=0.4$ and $z=4$ as follows.

1. Press “ $2^{\text{nd}}$ ” key and then “ $\text{DISTR}$” key.

  

Figure (1)

2. Go to “DRAW” and press “1.”

  

Figure (2)

3. Enter respective lower and upper $z$ values as shown below.

  

Figure (3)

4. Press “ENTER” to obtain the result as shown below.

  

Figure (4)

The probability is $0.3445$ .

Therefore, the required probability is $0.3445$ , and graph is shown in figure (4).

(b)

To determine

To find: The probability for $P\left(1<X<3\right)$ , and then sketch the area under curve.

Answer to Problem 8CYU

The required probability is $0.937$ , and graph is shown in figure (8).

Explanation of Solution

Given:

The number of sample is $925$ .

The mean is $1.8$ .

The standard deviation is $0.5$ .

Calculation:

Calculate the corresponding $z$ value for $X=1$ as follows.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{1-1.8}{0.5}\\ =-1.6\end{array}$

Calculate the corresponding $z$ value for $X=3$ as follows.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{3-1.8}{0.5}\\ =2.4\end{array}$

Calculate the required probability by using TI-83 calculator, and sketch the area between $z=-1.6$ and $z=2.4$ as follows.

5. Press “ $2^{\text{nd}}$ ” key and then “ $\text{DISTR}$” key.

  

Figure (5)

6. Go to “DRAW” and press “1.”

  

Figure (6)

7. Enter respective lower and upper $z$ values as shown below.

  

Figure (7)

8. Press “ENTER” to obtain the result as shown below.

  

Figure (8)

The probability is $0.937$ .

Therefore, the required probability is $0.937$ , and graph is shown in figure (8).