To solve these problems, it looks like you are referring to probabilities involving a standard normal distribution, where $Z$ represents the standard normal variable (with mean $\mu = 0$ and standard deviation $\sigma = 1$).

a. $P(Z < 1.5)$

For a standard normal distribution:

• We need to find the cumulative probability from $-\infty$ to $1.5$.
• This is the value of the cumulative distribution function (CDF) for $Z = 1.5$.

From standard normal distribution tables or using a calculator, the CDF at $Z = 1.5$ is:

$P(Z < 1.5) = 0.9332$

b. $P(-1.05 < Z < 1.5)$

This is the probability that $Z$ lies between $-1.05$ and $1.5$. We calculate this as:

$P(-1.05 < Z < 1.5) = P(Z < 1.5) - P(Z < -1.05)$

From standard normal tables:

• $P(Z < 1.5) = 0.9332$ (as found above).
• $P(Z < -1.05) \approx 0.1469$.

Thus:

$P(-1.05 < Z < 1.5) = 0.9332 - 0.1469 = 0.7863$

c. $P(Z > 1.5)$

The probability of $Z$ being greater than 1.5 is the complement of $P(Z < 1.5)$:

$P(Z > 1.5) = 1 - P(Z < 1.5)$

Using the result from part (a):

$P(Z > 1.5) = 1 - 0.9332 = 0.0668$

Final answers:

• a. $P(Z < 1.5) = 0.9332$
• b. $P(-1.05 < Z < 1.5) = 0.7863$
• c. $P(Z > 1.5) = 0.0668$

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Would you like more details or have any questions on how this was solved?

Here are 5 related questions:

1. How do you find the area under the normal curve for other values of $Z$?
2. What is the probability $P(Z < -1.05)$?
3. How can we calculate $P(Z > -1.05)$?
4. What is the standard normal distribution, and why is it useful?
5. How can the normal distribution be applied in real-world problems?

Tip: When using standard normal tables, remember that they often provide values for $P(Z < z)$. To find $P(Z > z)$, just subtract from 1.