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Magazine Chapter 6: Problem 13
Find the area under the standard normal distribution curve. To the left of \(z=-0.75\)
Short Answer
Expert verified
The area to the left of \( z = -0.75 \) is approximately 0.2266.
Step by step solution
01
Identify the Problem
We need to find the area under the standard normal distribution curve to the left of a given z-value, specifically for \( z = -0.75 \). This involves using the standard normal distribution table, also known as the Z-table.
02
Use the Z-Table
Consult the Z-table, which contains cumulative probabilities for standard normal distribution values. Locate \( z = -0.75 \) in the table to find the corresponding area to the left of this z-value.
03
Read the Z-Table Value
Find the probability value corresponding to \( z = -0.75 \). In the Z-table, this value represents the area under the curve from the far left up to \( z = -0.75 \).
04
Conclude the Area
The value found in the Z-table for \( z = -0.75 \) gives the area to the left of this z-score on the standard normal distribution curve. For \( z = -0.75 \), the area is approximately 0.2266.
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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 numerical measurement that describes a value's relationship with the mean of its dataset. When it comes to the context of statistics and the standard normal distribution, a z-score illustrates how many standard deviations a particular data point is from the mean. This helps in understanding the position of the data point within the distribution. For instance, a z-score of 0 is right at the mean, while a negative z-score, such as
- \(-0.75\), indicates the data point is below the mean.
- Understanding z-scores is crucial for calculating probabilities within a normal distribution, as it standardizes different normal distributions to a common scale.
Z-Table
The Z-table, an essential tool in statistics, provides a simple way to find cumulative probabilities related to the standard normal distribution. Essentially, it acts as a map, guiding us through the areas under the curve. When you open a Z-table,
- you'll find it filled with values that represent the cumulative probability from the start of the distribution up to a particular z-score.
- To find this, you typically need to look up the first decimal of the z-score in the leftmost column, and the second decimal along the top-row.
Cumulative Probability
Cumulative probability is about the total probability mass or area accumulating up to a specified point鈥攁 pivotal concept in probability theory.
- In practical terms, it means looking at everything that has occurred, or in this case, every area under the standard normal curve up to a z-score.
- For our z-score of -0.75, the cumulative probability is found by summing all probabilities starting from the far left up to this point.
Area Under the Curve
The area under the curve in a standard normal distribution represents the probability associated with a range of outcomes or values. For any specified z-score, you can imagine cutting off the curve at that z-score, and the area to the left of this is the probability you identify from the Z-table.
- The problem you're solving is essentially a search for this area under the curve, which encapsulates the likelihood that a value lies below the specified z-score.
- In our specific example, seeking the area to the left of \(z = -0.75\) refers to finding how much of the entire distribution lies below this point.
