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Chapter 12: Problem 11

In Exercises 9-16, find the percentage of data items in a normal distribution that lie between \(z=1\) and \(z=3\).

Short Answer

Expert verified
The percentage of data items in a normal distribution that lies between \(z = 1\) and \(z = 3\) is 15.74%.

Step by step solution

01

Understand the Z-Scores

The z-scores \(z = 1\) and \(z = 3\) represent standard deviations from the mean in a normal distribution. A z-score of 1 implies a value that is one standard deviation above the mean, while a z-score of 3 represents a value three standard deviations above the mean.
02

Use of the Z-Table

The next step is to use a standard normal distribution table, often referred to as the Z-table. This table shows the area to the left of any given z-score. When a z-score is known, this table can be used to find the proportion of the data set that is to the left of that z-score.
03

Calculation of the percentages

From the standard normal distribution table, look for the z value of 1 and find the corresponding probability. This probability is 0.8413. Now, look for the z value of 3 which has the corresponding probability of 0.9987. To find our required percentage, subtract the probability at \(z = 1\) from the probability at \(z = 3\). \(0.9987 - 0.8413 = 0.1574\). Interpreting this value as a percentage gives 15.74%. So, the percentage of the data items falling between z-scores 1 and 3 is 15.74 percent.
04

Interpret the Result

The result of this computation is interpreted as the percentage of all data in the normal distribution that falls between 1 standard deviation and 3 standard deviations above the mean, which is 15.74 percent.

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Key Concepts

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

Understanding Z-Scores
A z-score tells us how many standard deviations away a particular value is from the mean in a normal distribution. This measure helps us determine the position of a specific data point within the distribution. If the z-score is positive, it indicates that the data point is above the mean. Conversely, a negative z-score signifies that it is below the mean.
For example:
  • A z-score of 1 means the data point is one standard deviation above the mean.
  • A z-score of 3 signifies the data point is three standard deviations above the mean.
Understanding z-scores is crucial because they provide a standardized way for us to interpret and compare different data points in relation to the overall distribution.
The Role of Standard Deviations
Standard deviation is a statistical metric that reflects the amount of variation or dispersion in a set of data values. In a normal distribution, the mean represents the central value, while the standard deviation measures the extent of deviation from this mean.
Here's what you need to know:
  • A smaller standard deviation indicates that the data points are close to the mean.
  • A larger standard deviation shows that the data points are spread out over a wider range of values.
In the context of the exercise, each z-score directly corresponds to a number of standard deviations. For instance, a z-score of 1 represents one standard deviation away from the mean. Understanding standard deviations is essential for grasping how data varies within a distribution.
Using the Z-Table for Calculations
The Z-table, or standard normal distribution table, is a tool used to find the area under the curve to the left of a specific z-score in a standard normal distribution. This area represents the probability or proportion of data points that lie below that z-score.
Here's how you can use the Z-table:
  • To find the probability for a particular z-score, locate the z-score on the table.
  • Read the corresponding probability, which tells you how much of the data lies to the left of that z-score.
For the given exercise, you use the Z-table to find the probabilities at z-scores of 1 and 3, which help in determining that 15.74% of the data lies between these two z-scores when subtracted.
Probability in Statistics
Probability in statistics refers to the likelihood or chance of a particular outcome occurring within a defined set of parameters. It's a foundational concept in understanding how data behaves in various distributions, such as the normal distribution.
Key points about probability include:
  • It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
  • In a normal distribution, probabilities help us determine the percentage of data within certain intervals.
By using z-scores and the Z-table, we calculate the probability of data falling within specified intervals. In the exercise, this involved determining the likelihood that data would fall between z-scores of 1 and 3, leading to the conclusion that 15.74% of the data lies within this range.

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Most popular questions from this chapter

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