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Chapter 8: Problem 8

(A)\( make a sketch of the area under the standard normal curve corresponding to the probability and (b) find the value of the probability of the standard normal random variable \)Z$ corresponding to this area. $$ P(Z>2.24) $$

Short Answer

Expert verified
To sketch the standard normal curve, draw a symmetric bell-shaped curve with the mean at 0 and a standard deviation of 1. Shade the area to the right of \(Z = 2.24\) to represent \(P(Z>2.24)\). Using a standard normal distribution table or a calculator with a normal distribution function, find the area to the right of \(Z = 2.24\) under the standard normal curve. The probability \(P(Z > 2.24)\) is approximately 0.0125.

Step by step solution

01

Sketch the standard normal curve

To sketch the standard normal curve, draw a symmetric bell-shaped curve with the mean at 0 and the standard deviation of 1. The curve shows the probabilities associated with each value of the standard normal random variable \(Z\). Shade the area to the right of \(Z = 2.24\) to graphically represent \(P(Z>2.24)\).
02

Find the probability \(P(Z > 2.24)\)

To find the probability \(P(Z > 2.24)\), we can use either the standard normal distribution table or a calculator with a normal distribution function. In both cases, we need to find the area to the right of \(Z = 2.24\) under the standard normal curve. Using a standard normal distribution table: 1. First, find the area to the left of \(Z = 2.24\). Look for the row corresponding to 2.2 and the column corresponding to 0.04. The intersection will give the area to the left of 2.24. 2. The area to the left of \(Z = 2.24\) is approximately 0.9875. 3. To find the area to the right of \(Z = 2.24\), subtract the area to the left from 1: \(P(Z > 2.24) = 1 - 0.9875 = 0.0125\). Using a calculator with a normal distribution function: 1. Input the value \(Z = 2.24\) into the calculator's function for the standard normal distribution (usually denoted by \(NORMSDIST\) or similar). 2. The calculator will give the area to the left of \(Z = 2.24\). 3. To find the area to the right of \(Z = 2.24\), subtract the area to the left from 1. Both methods should give a probability of approximately 0.0125 for \(P(Z > 2.24)\).

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics which measures the likelihood of an event's occurrence. It is expressed as a number between 0 and 1. A probability of 0 indicates an impossible event, whereas a probability of 1 signifies a certain event. In the context of the standard normal distribution, probability is used to determine the likelihood that a particular value or range of values will occur.

In the exercise given, the task was to find the probability that the standard normal random variable, denoted as \(Z\), is greater than 2.24. This is symbolized as \(P(Z > 2.24)\). Understanding probability helps us predict outcomes in various situations, from academic research to real-world decision making.

When solving problems involving standard normal distributions, identifying the probability associated with a specific \(Z\)-score is essential. These probabilities tell us how often we can expect to see a \(Z\) value greater than a certain number, guiding our predictions and decision-making processes.
Standard Normal Curve
The standard normal curve, also known as the bell curve, is a symmetric, bell-shaped graph that represents a normal distribution. This curve is defined by its mean (0) and standard deviation (1) in the case of the standard normal distribution. Its shape ensures that most values cluster around the mean, with fewer values appearing as you move further away from it.

A crucial aspect of the curve is that it helps visualize the distribution of data. In the given exercise, drawing a sketch of the standard normal curve aids in illustrating the problem. It shows which area corresponds to the probability we are trying to find, highlighting the region to the right of \(Z = 2.24\).

Understanding the standard normal curve visually can make it easier to comprehend probabilities and areas under the curve. This graphical representation ensures that statistical results are more tangible and understandable. It enables us to see how rare or common particular outcomes are, based on their location on the curve.
Z-Score
A \(Z\)-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It's computed by taking the difference between the value and the mean, and then dividing that by the standard deviation. In a standard normal distribution:
  • The mean is 0
  • The standard deviation is 1
  • A \(Z\)-score tells us how many standard deviations away from the mean a point is
This concept is pivotal in the exercise as we are working with \(Z\)-scores to determine probabilities. In our case, the \(Z\)-score \(2.24\) tells us how far the point is from the average \(Z\)-score (which is 0 in a standard normal distribution).

The \(Z\)-score helps standardize different data points, making it possible to compare them directly. When applied to the standard normal distribution, \(Z\)-scores are used to find the corresponding probabilities in the normal distribution table, which illustrates the likelihood of these values occurring.
Normal Distribution Table
The normal distribution table, or Z-table, is a crucial tool in statistics for finding the probability that a standard normal random variable \(Z\) is less than or equal to a given value. It provides the cumulative distribution function values for \(Z\)-scores.

In the given exercise, the table was used to identify the area to the left of \(Z = 2.24\). Here's how it works:
  • Locate the row for 2.2 and the column for 0.04, intersecting at the probability value for \(Z = 2.24\).
  • Then, subtract this probability from 1 to find \(P(Z > 2.24)\).
This method is widely used because it simplifies the process of finding probabilities of standard normal distributions. Each entry in the table corresponds to the probability that a standard normal random variable is less than the given \(Z\)-score. Understanding how to use this table is essential for effectively solving standard normal distribution problems.

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