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Magazine Chapter 6: Problem 38
Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(z \geq-1.50) $$
Short Answer
Expert verified
The probability is approximately 0.9332.
Step by step solution
01
Understand the Standard Normal Distribution
The standard normal distribution is a bell-shaped curve where the mean is 0 and the standard deviation is 1. Random variable \( z \) under this distribution represents standard scores, often referred to as \( z \)-scores.
02
Identify the Probability Calculation
We are tasked with finding \( P(z \geq -1.50) \). This means we need to find the area under the standard normal curve to the right of \( z = -1.50 \).
03
Use Z-Table for Calculation
The standard normal distribution table (Z-table) provides the area to the left of a given \( z \)-value. First, find the area to the left of \( z = -1.50 \) using the Z-table.
04
Calculate Left Area from the Z-Table
Looking up \( z = -1.50 \) in the Z-table, we find the area to the left of \( z = -1.50 \), which is approximately 0.0668.
05
Determine the Complement Probability
Since \( P(z \geq -1.50) \) is the area to the right, calculate it using complement rule: \( P(z \geq -1.50) = 1 - P(z < -1.50) = 1 - 0.0668 = 0.9332 \).
06
Shade the Appropriate Area
On the standard normal curve, shade the region to the right of \( z = -1.50 \), which corresponds to the probability calculated as 0.9332.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Z-Score
The Z-Score is a crucial concept when working with the standard normal distribution. It represents the number of standard deviations a data point is from the mean.
Understanding how to compute and interpret the Z-score is essential for analyzing data within a normal distribution.
For a random variable with a standard normal distribution, the mean is 0, and the standard deviation is 1.
For a random variable with a standard normal distribution, the mean is 0, and the standard deviation is 1.
- A Z-score of 0 indicates that the data point is exactly at the mean.
- Positive Z-scores indicate the data point is above the mean, while negative Z-scores mean it's below the mean.
Probability Calculation
Probability calculation in the context of the standard normal distribution involves finding the area under the curve. For a given Z-score, this probability represents the likelihood of a random variable falling at or below that value.
In the problem, the goal is to find the probability that the Z is greater than or equal to -1.50 (i.e., the area to the right of -1.50 on the curve). Understanding how to interpret this probability aids in making decisions based on statistical data. The probability can give insights into how 'normal' or 'extreme' a particular observation is within a dataset. Remember:
In the problem, the goal is to find the probability that the Z is greater than or equal to -1.50 (i.e., the area to the right of -1.50 on the curve). Understanding how to interpret this probability aids in making decisions based on statistical data. The probability can give insights into how 'normal' or 'extreme' a particular observation is within a dataset. Remember:
- A larger area indicates a higher probability.
- Probabilities in the standard normal distribution are cumulative, meaning they start from the far left of the curve moving towards the right.
Z-Table Use
A Z-table, or standard normal distribution table, is a tool used to find the area (probability) to the left of a specific Z-score.
This table is essential for probability calculations involving the standard normal distribution.
In our case, we look up -1.50 in the Z-table to find the area to the left of this Z-score. When you find -1.50 in the table, you see an associated value of approximately 0.0668. This means the probability of a score being less than -1.50 is 6.68%. The Z-table is pivotal for:
In our case, we look up -1.50 in the Z-table to find the area to the left of this Z-score. When you find -1.50 in the table, you see an associated value of approximately 0.0668. This means the probability of a score being less than -1.50 is 6.68%. The Z-table is pivotal for:
- Quickly identifying the cumulative probability up to a Z-score.
- Serving as a basis to calculate complementary probabilities when needed.
Complement Rule
The Complement Rule is a fundamental principle in probability that helps simplify calculations. It states that the probability of an event occurring is equal to 1 minus the probability of it not occurring.
Applying this principle, we calculated the probability of Z being greater than -1.50. To use the complement rule:
Applying this principle, we calculated the probability of Z being greater than -1.50. To use the complement rule:
- First, find the area to the left using the Z-table, then subtract this from 1 to find the complement.
- For our problem: \[ P(z \geq -1.50) = 1 - P(z < -1.50) = 1 - 0.0668 = 0.9332 \]
