Questions: Determine the area under the standard normal curve that lies between (a) Z=-0.97 and Z=0.97, (b) Z=-1.93 and Z=0, and (c) Z=-1.38 and Z=-0.88.

Transcript text: Determine the area under the standard normal curve that lies between (a) $Z=-0.97$ and $Z=0.97$, (b) $Z=-1.93$ and $Z=0$, and (c) $Z=-1.38$ and $Z=-0.88$.

Solution

Solution Steps

Step 1: Area Between $ Z = -0.97 $ and $ Z = 0.97 $

To find the area under the standard normal curve between $ Z = -0.97 $ and $ Z = 0.97 $, we use the cumulative distribution function $ \Phi $:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.97) - \Phi(-0.97) \]

From the calculations, we find:

\[ P = 0.668 \]

Thus, the area that lies between $ Z = -0.97 $ and $ Z = 0.97 $ is $ 0.668 $.

Step 2: Area Between $ Z = -1.93 $ and $ Z = 0 $

Next, we calculate the area between $ Z = -1.93 $ and $ Z = 0 $:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0) - \Phi(-1.93) \]

The result of this calculation is:

\[ P = 0.4732 \]

Therefore, the area that lies between $ Z = -1.93 $ and $ Z = 0 $ is $ 0.4732 $.

Step 3: Area Between $ Z = -1.38 $ and $ Z = -0.88 $

Finally, we determine the area between $ Z = -1.38 $ and $ Z = -0.88 $:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.88) - \Phi(-1.38) \]

The calculated area is:

\[ P = 0.1056 \]

Thus, the area that lies between $ Z = -1.38 $ and $ Z = -0.88 $ is $ 0.1056 $.

Final Answer

(a) The area that lies between $ Z = -0.97 $ and $ Z = 0.97 $ is $ 0.668 $.
(b) The area that lies between $ Z = -1.93 $ and $ Z = 0 $ is $ 0.4732 $.
(c) The area that lies between $ Z = -1.38 $ and $ Z = -0.88 $ is $ 0.1056 $.

\[ \boxed{ \begin{align_} (a) & \quad 0.668 \\ (b) & \quad 0.4732 \\ (c) & \quad 0.1056 \end{align_} } \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!