Question

It is given that \(\text{Prob}(-1 \leq z \leq 1)=0.683\), \(\text{Prob}(-2 \leq z \leq 2)=0.954\), and \(\text{Prob}(-3 \leq z \leq 3)=0.997\), when \(z\) follows a standard normal distribution. If \(X\) follows a normal distribution with mean and variance as \(5\) and \(4\), respectively, then \(\text{Prob}(1 \leq X \leq 7)=\underline{}\) (rounded off to three decimal places).

Hint:

For a normal distribution, first convert \(X\) into \(z\) using \(z=\frac{X-\mu}{\sigma}\), then use standard normal probabilities and symmetry.

Answer 1

Correct Answer: 0.819

Step 1: Identify the mean and standard deviation.
It is given that \(X\) follows a normal distribution with mean \(5\) and variance \(4\).
Therefore,
\[ \mu=5 \] and
\[ \sigma^2=4 \]
So, the standard deviation is
\[ \sigma=\sqrt{4}=2 \]

Step 2: Convert \(X\) into standard normal variable \(z\).
The standard normal variable is given by
\[ z=\frac{X-\mu}{\sigma} \]
We need to find
\[ \text{Prob}(1 \leq X \leq 7) \]

Step 3: Convert the lower limit.
For \(X=1\),
\[ z=\frac{1-5}{2} \] \[ z=\frac{-4}{2} \] \[ z=-2 \]

Step 4: Convert the upper limit.
For \(X=7\),
\[ z=\frac{7-5}{2} \] \[ z=\frac{2}{2} \] \[ z=1 \]
Thus,
\[ \text{Prob}(1 \leq X \leq 7) = \text{Prob}(-2 \leq z \leq 1) \]

Step 5: Use the given standard normal probabilities.
We are given that
\[ \text{Prob}(-2 \leq z \leq 2)=0.954 \]
By symmetry of the standard normal distribution,
\[ \text{Prob}(-2 \leq z \leq 0)=\frac{0.954}{2}=0.477 \]
Also,
\[ \text{Prob}(-1 \leq z \leq 1)=0.683 \]
Again by symmetry,
\[ \text{Prob}(0 \leq z \leq 1)=\frac{0.683}{2}=0.3415 \]

Step 6: Add the required probability parts.
Now,
\[ \text{Prob}(-2 \leq z \leq 1) = \text{Prob}(-2 \leq z \leq 0)+\text{Prob}(0 \leq z \leq 1) \]
\[ \text{Prob}(-2 \leq z \leq 1) = 0.477+0.3415 \] \[ \text{Prob}(-2 \leq z \leq 1) = 0.8185 \]
Rounded off to three decimal places,
\[ 0.8185 \approx 0.819 \]
Therefore,
\[ \boxed{0.819} \]