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.