Let   find P(-1.34 < Z < 2.37).

Give answer correct to 4dp.

To find the probability \( P(-1.34 < Z < 2.37) \), where \( Z \) is a standard normal variable (with a mean of 0 and a standard deviation of 1), we can use the standard normal distribution table or a calculator.

We proceed by finding:

1. \( P(Z < 2.37) \)
2. \( P(Z < -1.34) \)

Then, we subtract the two probabilities:

\[
P(-1.34 < Z < 2.37) = P(Z < 2.37) - P(Z < -1.34)
\]

**Step 1: Find \( P(Z < 2.37) \):**

Using the standard normal distribution table:

\[
P(Z < 2.37) \approx 0.9911
\]

**Step 2: Find \( P(Z < -1.34) \):**

Using the standard normal distribution table:

\[
P(Z < -1.34) \approx 0.0901
\]

**Step 3: Subtract the two probabilities:**

\[
P(-1.34 < Z < 2.37) = 0.9911 - 0.0901 = 0.9010
\]

Thus, \( P(-1.34 < Z < 2.37) \approx 0.9010 \) (correct to four decimal places).

Would you like any further details on this calculation, or do you have other questions?

Here are 5 related questions for further exploration:
1. How do you find \( P(-2 < Z < 1.5) \) using the same method?
2. What is the probability \( P(Z > 1.34) \) for the standard normal distribution?
3. How do you interpret the result of \( P(-1.34 < Z < 2.37) \) in terms of percentage?
4. How would the probability change if the interval was \( P(-1.34 < Z < 1.5) \)?
5. What is the value of \( Z \) for which \( P(Z < z) = 0.9010 \)?

**Tip:** Always ensure that you are reading values from the correct side of the standard normal distribution table. This avoids common errors, especially with negative \( Z \)-scores.

Learn how to calculate the probability P(-1.34 < Z < 2.37) using the standard normal distribution. Follow a step-by-step solution including finding probabilities for specific Z-values and subtracting them. Suitable for advanced high school students and beyond.

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