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Chapter 6: Problem 17

Find the area under the standard normal distribution curve. $$ \text { Between } z=-1.46 \text { and } z=-1.77 $$

Short Answer

Expert verified
The area between z = -1.46 and z = -1.77 is 0.0337.

Step by step solution

01

Understand the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The area under the curve between two points corresponds to the probability of a random variable falling between those points.
02

Use Z-table to Find Probabilities

Locate the z-values -1.46 and -1.77 in the Z-table. These values represent the area from the left end of the curve to each z-value. Find the probability for each z-value. For z = -1.46, the Z-table shows P(Z < -1.46) = 0.0721. For z = -1.77, the Z-table shows P(Z < -1.77) = 0.0384.
03

Calculate the Area Between the Z-values

To find the area between z = -1.46 and z = -1.77, subtract the smaller area from the larger area. This results in 0.0721 - 0.0384 = 0.0337.

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Key Concepts

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

Understanding Z-values
Z-values, also known simply as z-scores, are a way to standardize scores on a normal distribution. They represent the number of standard deviations a given data point is from the mean. In the case of a standard normal distribution, the mean is 0 and the standard deviation is 1.

To calculate a z-value for a given data point, you use the formula: \[ z = \frac{(X - \mu)}{\sigma} \]Where:
  • X is the value of the data point.
  • \( \mu \) is the mean of the distribution.
  • \( \sigma \) is the standard deviation.
Z-values help us understand how unusual or typical a given score is within a set of data. They are crucial for calculating probabilities in a normal distribution.
Calculating Probability in the Standard Normal Distribution
Probability refers to the likelihood of a particular event occurring. In terms of the standard normal distribution, it represents the area under the curve between two z-values.

When working with the standard normal distribution:
  • The total area under the curve equals 1, or 100% probability.
  • You can find probabilities by locating the relevant area under the curve.
To calculate probabilities between two z-values, like between -1.46 and -1.77, you first find the area up to each z-value using a Z-table. Then, you subtract the smaller area from the larger one to get the probability for the range.

This method allows you to determine how likely it is for a random variable to land between two specific points on the distribution.
Utilizing the Z-table
The Z-table, also known as the standard normal table, is a key tool in statistics for finding the probability associated with a given z-score. This table lists the probability of a standard normal variable (Z) being less than a specific value. To use a Z-table effectively:
  • Find the row associated with the z-value鈥檚 integer and first decimal (e.g., for z = -1.46, locate -1.4).
  • Next, find the column for the second decimal place (e.g., .06 for -1.46).
  • The intersection of the row and column gives the cumulative probability up to that z-value.
For instance, if P(Z < -1.46) = 0.0721 and P(Z < -1.77) = 0.0384 , the Z-table shows that about 7.21% of data falls below z = -1.46 and about 3.84% falls below z = -1.77.

By understanding and using the Z-table, you can quickly calculate probabilities and gain insights into your data, which is essential for data analysis.

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Most popular questions from this chapter

Explain why a normal distribution can be used as an approximation to a binomial distribution.

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