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 > 200) =

(b) P(z <= 1,44) =

(c) P(- 1.09 < Z < 220) =

To solve these problems, we use the properties of the standard normal distribution, where \( Z \) follows a normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). We can find the probabilities using a standard normal table or a calculator. Below are the answers:

### (a) \( P(Z > 200) \)
- Since \( Z = 200 \) is far beyond the typical range of a standard normal distribution, the probability that \( Z \) is greater than 200 is extremely small, essentially zero.
  
  \[
  P(Z > 200) \approx 0
  \]

### (b) \( P(Z \leq 1.44) \)
- To find this, we use the cumulative distribution function (CDF) for the standard normal distribution at \( Z = 1.44 \).

  \[
  P(Z \leq 1.44) \approx 0.925
  \]

### (c) \( P(-1.09 < Z < 220) \)
- First, \( P(Z < 220) \) is extremely close to 1, so we focus on the range from \( -1.09 \) to \( 220 \).

  \[
  P(-1.09 < Z < 220) \approx P(Z > -1.09) \approx 1 - P(Z \leq -1.09)
  \]
  Using the CDF for \( Z = -1.09 \):
  \[
  P(Z \leq -1.09) \approx 0.1379
  \]
  So,
  \[
  P(-1.09 < Z < 220) \approx 1 - 0.1379 = 0.8621
  \]

Let me know if you want details on these calculations or have any questions!

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Here are 8 related questions you might want to explore:
1. What is \( P(Z > 1.96) \)?
2. How do you calculate \( P(Z < -2.33) \)?
3. What is the 95th percentile of the standard normal distribution?
4. How do you interpret the z-score in real-world terms?
5. What is \( P(-0.5 < Z < 0.5) \)?
6. How do you find \( P(Z > 0.84) \)?
7. How does the standard normal distribution differ from other normal distributions?
8. How do you convert a normal variable to a standard normal variable?

**Tip**: The cumulative distribution function (CDF) is crucial for finding probabilities in a normal distribution, giving the area under the curve to the left of a specific z-score.

Learn how to calculate probabilities using the standard normal distribution. Explore solutions for specific queries involving probabilities such as P(z > 200), P(z <= 1.44), and P(-1.09 < Z < 220).

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