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 > 2.00) =

(b) P(z <= 1.44) =

(c) P(- 1.09 < Z < 2.20) =

Let's calculate each of the probabilities based on the standard normal distribution \( Z \), where \( Z \) has a mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). We'll use the cumulative distribution function (CDF) of the standard normal distribution.

### (a) \( P(Z > 2.00) \)
- To find this probability, we calculate \( P(Z > 2.00) \). Using the CDF for \( Z = 2.00 \), we find the area to the left of 2.00 and subtract it from 1.
  
  \[
  P(Z \leq 2.00) \approx 0.9772
  \]
  Therefore,
  \[
  P(Z > 2.00) = 1 - 0.9772 = 0.0228
  \]

  **Answer**: \( P(Z > 2.00) \approx 0.023 \)

### (b) \( P(Z \leq 1.44) \)
- Here, we need the cumulative probability for \( Z \leq 1.44 \).
  
  \[
  P(Z \leq 1.44) \approx 0.9251
  \]

  **Answer**: \( P(Z \leq 1.44) \approx 0.925 \)

### (c) \( P(-1.09 < Z < 2.20) \)
- This is the probability that \( Z \) lies between \( -1.09 \) and \( 2.20 \). We calculate:
  
  \[
  P(Z \leq 2.20) \approx 0.9861
  \]
  \[
  P(Z \leq -1.09) \approx 0.1379
  \]
  Therefore,
  \[
  P(-1.09 < Z < 2.20) = 0.9861 - 0.1379 = 0.8482
  \]

  **Answer**: \( P(-1.09 < Z < 2.20) \approx 0.848 \)

Let me know if you'd like further details on these calculations or have any questions!

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Here are 8 related questions you might find interesting:
1. What is \( P(Z > 1.96) \)?
2. How do you calculate \( P(Z < -1.64) \)?
3. What is the probability that \( Z \) lies between 0 and 2.33?
4. How do you interpret a z-score of 3.0 in terms of probability?
5. What is \( P(-0.5 < Z < 0.5) \)?
6. How do you find the area under the curve to the right of \( Z = -1.5 \)?
7. What is the 90th percentile of the standard normal distribution?
8. How do you standardize a variable to calculate probabilities?

**Tip**: Always remember that the total area under the standard normal curve equals 1, so probabilities for values outside a given range can often be easily calculated by subtracting from 1.

Learn how to calculate probabilities using the standard normal distribution, focusing on probabilities such as P(z > 2.00), P(z <= 1.44), and P(-1.09 < Z < 2.20). Detailed step-by-step solutions provided with rounded answers to three decimal places.

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