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Chapter 6: Problem 33

Find the probabilities for each, using the standard normal distribution. \(P(z< -1.51)\)

Short Answer

Expert verified
The probability \(P(z < -1.51)\) is approximately 0.0655.

Step by step solution

01

Understand the Standard Normal Distribution

The standard normal distribution, also known as the Z-distribution, is a normal distribution with a mean of 0 and a standard deviation of 1. We use the Z-table to find probabilities associated with z-scores.
02

Interpret the Problem

The problem asks for the probability that the z-score is less than -1.51. This is denoted as \(P(z < -1.51)\).
03

Use the Z-table

Locate the z-score of -1.51 in the Z-table. The Z-table gives us the cumulative probability from the far left up to the z-score. The standard Z-table shows probabilities for z-scores less than a specific value, which matches our requirement.
04

Read the Z-table Value

In the Z-table, find the row for -1.5 and the column for 0.01. The intersection of these gives the probability \(P(z < -1.51)\). This value is approximately 0.0655.
05

Record the Probability

The probability \(P(z < -1.51)\) is approximately 0.0655. This means that approximately 6.55% of the data falls below a z-score of -1.51 in a standard normal distribution.

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Key Concepts

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

Understanding the Z-Score
The z-score is a crucial concept in statistics, representing the number of standard deviations a data point is from the mean. In the context of the standard normal distribution, a z-score allows us to understand how extreme a value is compared to the average. Whenever we talk about probabilities in a standard normal distribution, we're essentially dealing with z-scores.
  • A positive z-score indicates a value above the mean.
  • A negative z-score indicates a value below the mean.
  • Z-scores are dimensionless since they're derived from a ratio.
For instance, a z-score of -1.51 signals that the value is 1.51 standard deviations below the mean of the distribution, which is 0 in the standard normal distribution. By recognizing the z-score's position, we can then proceed to find its probability or proportion related to that position.
What is Cumulative Probability?
Cumulative probability is the likelihood that a random variable is less than or equal to a particular value. When using the standard normal distribution, this involves summing the probabilities from the left end of the distribution up to a specific z-score. This cumulative aspect allows us to determine how much of the data falls below a specific point.
  • It's useful for understanding the proportion of data below a certain value.
  • In our example, it helps us find the probability that a z-score is less than -1.51.
  • Cumulative probabilities range from 0 to 1.
When we calculate cumulative probabilities, we're gathering information on how data points accumulate as we move along the distribution. In essence, it's a cumulative sum on a continuous curve, representing the entire area under the curve from the far left until our z-score of interest.
Navigating the Z-Table
The Z-table is a fundamental tool in statistics for finding probabilities related to the standard normal distribution. It provides the cumulative probability of z-scores and is structured to be user-friendly once you understand how it works. To use the Z-table effectively:
  • Identify the z-score you're interested in (e.g., -1.51).
  • Decipher the table by locating the row and column corresponding to your z-score. Rows typically correspond to the z-score's whole and tenths鈥� parts, while columns represent the hundredths.
  • The intersection of these row and column gives the cumulative probability for that z-score.
For the z-score -1.51, you'd locate the -1.5 row of the Z-table and the 0.01 column. The value at the intersection gives the probability of the z-score being less than -1.51, which in our example is approximately 0.0655. This tool thus simplifies the process of finding cumulative probabilities, allowing us to deduct various statistical measures effortlessly.

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Most popular questions from this chapter

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