Suppose Z

 follows the standard normal distribution. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places.



(a)=

P

>

Z

−

1.93

(b)=

P

≤

Z

−

1.14

(c)=

P

<

0.87

<

Z

2.05

To solve the probability problems using the **standard normal distribution** (Z) and rounding to at least three decimal places, we will use standard normal distribution properties and cumulative probabilities:

### (a) \( P(Z > -1.93) \)

The probability that \( Z \) is greater than a certain value is equivalent to:

\[
P(Z > -1.93) = 1 - P(Z \leq -1.93)
\]

Looking up \( P(Z \leq -1.93) \) from the cumulative distribution table, we get approximately:

\[
P(Z \leq -1.93) \approx 0.0268
\]

So:

\[
P(Z > -1.93) = 1 - 0.0268 = 0.9732
\]

### (b) \( P(Z \leq -1.14) \)

Looking up the cumulative probability for \( Z \leq -1.14 \):

\[
P(Z \leq -1.14) \approx 0.1271
\]

### (c) \( P(0.87 < Z < 2.05) \)

This is the probability that \( Z \) lies between two values, and can be calculated as:

\[
P(0.87 < Z < 2.05) = P(Z < 2.05) - P(Z \leq 0.87)
\]

First, look up the cumulative probabilities:

\[
P(Z \leq 2.05) \approx 0.9798
\]
\[
P(Z \leq 0.87) \approx 0.8078
\]

Now subtract the two:

\[
P(0.87 < Z < 2.05) = 0.9798 - 0.8078 = 0.1720
\]

### Final answers:

(a) \( P(Z > -1.93) = 0.9732 \)

(b) \( P(Z \leq -1.14) = 0.1271 \)

(c) \( P(0.87 < Z < 2.05) = 0.1720 \)

---

Do you want more details on these calculations or have any questions?

Here are five related questions to further explore the topic:
1. What is the probability of \( Z \leq 0 \) in the standard normal distribution?
2. How do you compute \( P(Z > 1.5) \) in a standard normal distribution?
3. What is the symmetry property of the normal distribution that helps simplify these calculations?
4. How would you calculate \( P(-2 < Z < 1) \)?
5. Can you explain how the standard normal distribution is related to z-scores in real-world data?

**Tip:** Always remember that probabilities for \( P(Z > x) \) can be computed by subtracting the cumulative probability from 1.

Suppose Z follows the standard normal distribution. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places. (a) P(Z > -1.93) (b) P(Z ≤ -1.14) (c) P(0.87 < Z < 2.05)

Learn how to calculate probabilities using the standard normal distribution for Z scores. This guide covers solving P(Z > -1.93), P(Z ≤ -1.14), and P(0.87 < Z < 2.05) with detailed step-by-step solutions. Suitable for students in Grades 10-12.

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