Question

A Past Exam Question Random variable z follows the standard normal distribution. In each of the following, sketch the standard normal distribution to illustrate your result.

A.Find p(-3.0B.Find p(z>1.0)

Answer 1

Certainly, let's address each part:

A. To find \(P(Z < -3.0)\) in the standard normal distribution, you're looking for the area to the left of \(z = -3.0\). This area represents the cumulative probability up to \(z = -3.0\). The standard normal distribution is symmetric, so you can look up this value in a standard normal distribution table or use a calculator to find the cumulative probability.

B. For \(P(Z > 1.0)\), you're interested in the area to the right of \(z = 1.0\). Again, you can use a standard normal distribution table or a calculator to find this cumulative probability.

I'll describe the general approach for both cases. Keep in mind that the exact values would need to be looked up or calculated.

1. **For A:** Sketch the standard normal distribution curve. Shade the area to the left of \(z = -3.0\) to illustrate \(P(Z < -3.0)\).

2. **For B:** Sketch the standard normal distribution curve again. Shade the area to the right of \(z = 1.0\) to illustrate \(P(Z > 1.0)\).

Answer 2

The student's math question involves the use of the standard normal distribution to find probabilities corresponding to specific z-scores by using a z-table. The z-scores indicate the number of standard deviations a score is from the mean of the distribution.

Step-by-step explanation:

The question deals with the standard normal distribution where a random variable z follows a standard normal distribution, often represented as Z ~ N(0, 1). This implies that the mean of the distribution is 0 and the standard deviation is 1. To find probabilities related to standard normal distribution, we calculate the z-score and refer to a z-table for corresponding probabilities. The z-score tells us how many standard deviations our score is from the mean.

To address the specifics of the student's question:

• For p(z < -3.0), we would want to find the area to the left of z = -3.0 on the standard normal distribution curve, which represents the probability.
• For p(z > 1.0), we look for the area under the curve to the right of z = 1.0.

The probability is represented by the area under the curve. A z-score of -3.0 or 1.0 is quite extreme, as these scores are three and one standard deviation away from the mean, respectively.

When calculating such probabilities using a z-table, the table typically shows the area to the left of a given z-score. We can use the table directly to answer p(z < -3.0) or we might need to subtract the table value from one to find p(z > 1.0).