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 > - 0.81) =

(b) p(z <= - 1.65) =

(c) P(0.35 < Z < 1.75) =

Let's calculate the probabilities based on the standard normal distribution \( Z \), where \( Z \) has a mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). We'll determine these probabilities step by step:

### (a) \( P(Z > -0.81) \)
- To find this probability, we calculate the area to the right of \( Z = -0.81 \). Using the cumulative distribution function (CDF) for \( Z = -0.81 \):
  
  \[
  P(Z \leq -0.81) \approx 0.2090
  \]
  Therefore,
  \[
  P(Z > -0.81) = 1 - 0.2090 = 0.7910
  \]

  **Answer**: \( P(Z > -0.81) \approx 0.791 \)

### (b) \( P(Z \leq -1.65) \)
- Here, we find the cumulative probability for \( Z \leq -1.65 \):
  
  \[
  P(Z \leq -1.65) \approx 0.0495
  \]

  **Answer**: \( P(Z \leq -1.65) \approx 0.050 \)

### (c) \( P(0.35 < Z < 1.75) \)
- This is the probability that \( Z \) lies between \( 0.35 \) and \( 1.75 \). We calculate:
  
  \[
  P(Z \leq 1.75) \approx 0.9599
  \]
  \[
  P(Z \leq 0.35) \approx 0.6368
  \]
  Therefore,
  \[
  P(0.35 < Z < 1.75) = 0.9599 - 0.6368 = 0.3231
  \]

  **Answer**: \( P(0.35 < Z < 1.75) \approx 0.323 \)

Let me know if you'd like further explanations or have more questions!

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Here are 8 related questions you might find useful:
1. What is \( P(Z > 0.5) \)?
2. How do you calculate \( P(Z < -2.0) \)?
3. What is the probability that \( Z \) lies between -1 and 1?
4. How do you interpret a z-score of -0.75 in terms of probability?
5. What is \( P(-0.5 < Z < 1.0) \)?
6. How do you find the area under the curve to the left of \( Z = 0.25 \)?
7. What is the 95th percentile of the standard normal distribution?
8. How do you standardize a score using the z-score formula?

**Tip**: When dealing with standard normal distribution probabilities, using the symmetry of the distribution can simplify calculations, especially for negative z-scores.

Learn how to calculate probabilities using the standard normal distribution with step-by-step examples. Solve for P(z > -0.81), P(z <= -1.65), and P(0.35 < Z < 1.75) with detailed explanations and rounded responses to at least three decimal places.

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