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> 1.65)=

(b) P (Z≤ 0.74)=

(c) P(-108 <2<2.20)=

Let's calculate the probabilities using the standard normal distribution \( Z \) with a mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). We'll find these probabilities step by step:

### (a) \( P(Z > 1.65) \)
- To find this probability, we calculate the area to the right of \( Z = 1.65 \). Using the cumulative distribution function (CDF) for \( Z = 1.65 \):
  
  \[
  P(Z \leq 1.65) \approx 0.9505
  \]
  Therefore,
  \[
  P(Z > 1.65) = 1 - 0.9505 = 0.0495
  \]

  **Answer**: \( P(Z > 1.65) \approx 0.050 \)

### (b) \( P(Z \leq 0.74) \)
- Here, we find the cumulative probability for \( Z \leq 0.74 \):
  
  \[
  P(Z \leq 0.74) \approx 0.7704
  \]

  **Answer**: \( P(Z \leq 0.74) \approx 0.770 \)

### (c) \( P(-1.08 < Z < 2.20) \)
- This is the probability that \( Z \) lies between \( -1.08 \) and \( 2.20 \). We calculate:
  
  \[
  P(Z \leq 2.20) \approx 0.9861
  \]
  \[
  P(Z \leq -1.08) \approx 0.1401
  \]
  Therefore,
  \[
  P(-1.08 < Z < 2.20) = 0.9861 - 0.1401 = 0.8460
  \]

  **Answer**: \( P(-1.08 < Z < 2.20) \approx 0.846 \)

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

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Here are 8 related questions you might find useful:
1. What is \( P(Z > 1.96) \)?
2. How do you calculate \( P(Z < -1.28) \)?
3. What is the probability that \( Z \) lies between 0 and 1.96?
4. How do you interpret a z-score of -1.5 in terms of probability?
5. What is \( P(-2 < Z < 0) \)?
6. How do you find the area under the curve to the left of \( Z = 1.5 \)?
7. What is the 99th percentile of the standard normal distribution?
8. How do you convert raw scores to z-scores in a normal distribution?

**Tip**: When working with probabilities involving the standard normal distribution, always remember to use symmetry properties of the normal curve to simplify calculations.

Learn how to calculate probabilities using the standard normal distribution with detailed step-by-step solutions. Includes calculations for P(Z > 1.65), P(Z ≤ 0.74), and P(-1.08 < Z < 2.20) rounded to three decimal places.

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