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Chapter 7: Problem 44

Find the indicated probability of the standard normal random variable \(Z\). $$P(Z<-0.38 \text { or } Z>1.93)$$

Short Answer

Expert verified
P(Z < -0.38 or Z > 1.93) = 0.3785

Step by step solution

01

Understand the Problem

The exercise asks for the probability that a standard normal variable, denoted as Z, is either less than -0.38 or greater than 1.93. This needs to be represented as two separate probabilities combined together.
02

Use the Standard Normal Distribution Table

The standard normal distribution table provides the area (probability) to the left of a given Z-value. We will use this table to find the probabilities for Z < -0.38 and Z > 1.93.
03

Find Probability for Z < -0.38

Use the standard normal distribution table to find the probability that Z is less than -0.38. This is represented as P(Z < -0.38). From the table, P(Z < -0.38) 鈮� 0.3517.
04

Find Probability for Z > 1.93

Now, find the probability that Z is greater than 1.93. This is represented as P(Z > 1.93). Since the table gives the area to the left of the Z value, we first find P(Z < 1.93) from the table, which is approximately 0.9732. Therefore, P(Z > 1.93) = 1 - P(Z < 1.93) = 1 - 0.9732 = 0.0268.
05

Combine the Probabilities

The total probability is the sum of the probabilities found in Steps 3 and 4. Therefore, P(Z < -0.38 or Z > 1.93) = P(Z < -0.38) + P(Z > 1.93) = 0.3517 + 0.0268 = 0.3785.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability
Probability is a measure that quantifies the likelihood that an event will occur. It ranges from 0 to 1, where 0 indicates an impossible event and 1 indicates certainty. For the standard normal distribution, the total area under the curve is 1, representing 100% certainty. Breaking this down, different sections of the curve show the probability of the variable falling within that range. When calculating the probability for a given Z-value, we essentially find the area under the curve to the left of that Z-value.
What is a Z-Value?
A Z-value, or Z-score, represents the number of standard deviations a data point is from the mean. In the context of a standard normal distribution, the mean is zero and the standard deviation is one. A Z-value allows us to understand where a particular value lies within the context of a normal distribution. For example, a Z-value of -0.38 means that the point is 0.38 standard deviations below the mean.
Using the Normal Distribution Table
The normal distribution table provides the cumulative probability associated with a given Z-value in a standard normal distribution. Essentially, it tells us the probability that a value is less than a specified Z-value. For example, if we have a Z-value of -0.38, we can look this up in the table to find that P(Z < -0.38) 鈮� 0.3517. This means there is approximately a 35.17% chance that a value from this distribution is less than -0.38.
Combining Probabilities
In some problems, you may need to find the combined probabilities of multiple events. For our example, we want to know the probability that Z is either less than -0.38 or greater than 1.93. First, we found the individual probabilities:
  • P(Z < -0.38) 鈮� 0.3517
  • P(Z > 1.93) = 1 - P(Z < 1.93) = 1 - 0.9732 = 0.0268
Combining these, we simply add the probabilities:
  • P(Z < -0.38 or Z > 1.93) = 0.3517 + 0.0268 = 0.3785
Thus, there is a 37.85% chance that Z falls below -0.38 or above 1.93.

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Most popular questions from this chapter

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