The image shows two probability problems related to a standard normal distribution (z-distribution), each involving shading specific areas under the normal curve according to given probability statements.

Problem 1

The first problem asks to sketch the region corresponding to the statement: $P(-0.7 < z < -0.3)$ This specifies the probability that the z-score $z$ falls between -0.7 and -0.3. The appropriate shading is done between these two values on the standard normal curve, illustrating the area that represents the probability between -0.7 and -0.3.

Problem 2

The second problem asks to sketch the region corresponding to the statement: $P(z < c) = 0.45$ This statement specifies the cumulative probability that the z-score $z$ is less than some critical value $c$, where this probability equals 0.45. This implies that $c$ is located at approximately $-1.0$ on the z-scale (since $P(z < -1.0) \approx 0.45$ in a standard normal distribution). The shaded region to the left of -1.0 represents this probability.

Would you like further details on calculating probabilities in the standard normal distribution, or do you have specific questions?

Here are some related questions that may help deepen your understanding:

1. How is the area under the normal curve related to probability in standard normal distributions?
2. What steps are involved in finding the z-scores for given probabilities?
3. How can one calculate probabilities for non-standard normal distributions?
4. How does symmetry of the normal distribution help in solving these types of problems?
5. What is the relationship between cumulative probability and z-scores?

Tip: When working with standard normal distributions, using z-tables or technology (such as statistical software or calculators) can simplify finding cumulative probabilities and critical z-values.