Question
Let \( Z \sim N(0,1) \). Accurate to 4 decimal places, evaluate \( P(-0.37 \leq Z \leq-0.07) \) 0.3557 0.1164 0.8836 0.4721 None of the above.
Ask by Frank Gardner. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The probability \( P(-0.37 \leq Z \leq -0.07) \) is approximately 0.1164.
Solution
We are to evaluate
\[
P(-0.37 \leq Z \leq -0.07)
\]
for \( Z \sim N(0,1) \).
**Step 1. Express the probability in terms of the CDF**
The probability can be written as
\[
P(-0.37 \leq Z \leq -0.07) = \Phi(-0.07) - \Phi(-0.37)
\]
where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.
**Step 2. Use the symmetry property of the standard normal distribution**
Recall the property
\[
\Phi(-z) = 1 - \Phi(z).
\]
Using this:
\[
\Phi(-0.07) = 1 - \Phi(0.07)
\]
\[
\Phi(-0.37) = 1 - \Phi(0.37).
\]
**Step 3. Substitute and simplify**
Substitute in the original expression:
\[
P(-0.37 \leq Z \leq -0.07) = \bigl[1 - \Phi(0.07)\bigr] - \bigl[1 - \Phi(0.37)\bigr] = \Phi(0.37) - \Phi(0.07).
\]
**Step 4. Look up the standard normal values**
Using a standard normal table or calculator:
\[
\Phi(0.07) \approx 0.5279,
\]
\[
\Phi(0.37) \approx 0.6443.
\]
**Step 5. Compute the difference**
Now subtract:
\[
P(-0.37 \leq Z \leq -0.07) \approx 0.6443 - 0.5279 = 0.1164.
\]
Thus, accurate to 4 decimal places, the probability is \( \boxed{0.1164} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To find \( P(-0.37 \leq Z \leq -0.07) \), we can use the cumulative distribution function (CDF) for the standard normal distribution, which is represented as \( \Phi(z) \). First, we need to determine \( \Phi(-0.07) \) and \( \Phi(-0.37) \): 1. Using standard normal distribution tables or software: - \( \Phi(-0.07) \approx 0.4721 \) - \( \Phi(-0.37) \approx 0.3557 \) Now, we calculate the probability: \[ P(-0.37 \leq Z \leq -0.07) = \Phi(-0.07) - \Phi(-0.37) \] Substituting the values we found: \[ P(-0.37 \leq Z \leq -0.07) = 0.4721 - 0.3557 \approx 0.1164 \] Thus, the correct answer is **0.1164**.
