Suppose Z follows the standard normal, distribution. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places.

(0) p(z <= 172) =

(b) P(z > - 203) =

(x) F(0.9 < z < 2.02) =

To solve these problems, we can use the properties of the standard normal distribution \( Z \), which has a mean of 0 and a standard deviation of 1. The ALEKS calculator or any standard normal distribution table can be used for these calculations.

### (a) \( P(z \leq 172) \)
This is asking for the cumulative probability up to \( z = 172 \). However, \( z = 172 \) is far in the tail of the distribution, meaning that the cumulative probability is effectively 1 (or very close to it).

So:
\[ P(z \leq 172) \approx 1.000 \]

### (b) \( P(z > -203) \)
This is asking for the probability that \( z \) is greater than \(-203\). Since \(-203\) is also far in the tail but on the negative side, this effectively means the probability is nearly 1.

So:
\[ P(z > -203) \approx 1.000 \]

### (x) \( P(0.9 < z < 2.02) \)
This is asking for the probability that \( z \) is between 0.9 and 2.02. This can be calculated by finding the cumulative probability at 2.02 and subtracting the cumulative probability at 0.9.

Let's calculate:

1. \( P(z \leq 2.02) \)
2. \( P(z \leq 0.9) \)
3. Subtract these to get the final probability.

Using the standard normal distribution:
\[ P(z \leq 2.02) \approx 0.9783 \]
\[ P(z \leq 0.9) \approx 0.8159 \]

Now, subtract the two:
\[ P(0.9 < z < 2.02) = 0.9783 - 0.8159 = 0.1624 \]

So:
\[ P(0.9 < z < 2.02) \approx 0.162 \]

### Summary:
- \( P(z \leq 172) \approx 1.000 \)
- \( P(z > -203) \approx 1.000 \)
- \( P(0.9 < z < 2.02) \approx 0.162 \)

Would you like more details or have any further questions?

Here are some related questions:

1. What is the significance of the standard normal distribution?
2. How do you calculate probabilities for non-standard normal distributions?
3. What are z-scores, and how are they used in statistics?
4. How does the cumulative distribution function (CDF) relate to probability?
5. What are some real-world applications of the standard normal distribution?
6. How can one interpret a z-score in the context of probability?
7. What is the difference between a one-tailed and a two-tailed test?
8. How does one use the inverse standard normal distribution?

**Tip:** When working with standard normal distribution tables, remember that they typically provide cumulative probabilities from the left up to a specific z-value.

Learn how to calculate probabilities using the standard normal distribution with detailed examples. This page covers calculating cumulative probabilities and ranges of z-scores. Round your responses to at least three decimal places.

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