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.