To solve the problem $P(-0.1 < z < 0.84)$ for a standard normal distribution, we need to compute the cumulative probabilities corresponding to these z-values.

1. Find the cumulative probability for $z = 0.84$: Using the standard normal distribution table (Z-table), or a calculator, the cumulative probability for $z = 0.84$ is approximately: $P(z \leq 0.84) \approx 0.7995$

2. Find the cumulative probability for $z = -0.1$: Similarly, the cumulative probability for $z = -0.1$ is: $P(z \leq -0.1) \approx 0.4602$

3. Subtract the two cumulative probabilities: To find the probability that $z$ lies between -0.1 and 0.84, subtract the cumulative probability for $z = -0.1$ from the cumulative probability for $z = 0.84$: $P(-0.1 < z < 0.84) = P(z \leq 0.84) - P(z \leq -0.1)$ $P(-0.1 < z < 0.84) \approx 0.7995 - 0.4602 = 0.3393$

Thus, the probability $P(-0.1 < z < 0.84)$ is approximately 0.3393.

Would you like more details or explanations?

Here are 5 related questions for further exploration:

1. What is the probability that $z$ is greater than 0.84 in a standard normal distribution?
2. How would you find $P(z < -0.1)$ in this scenario?
3. What is the total area under the standard normal curve?
4. How do you interpret probabilities in the context of a normal distribution?
5. How can this result be used in hypothesis testing?

Tip: For quick standard normal distribution calculations, many scientific calculators have built-in functions that can compute the cumulative distribution function (CDF) values directly.