Problem 14 For Exercises 7 through $26,$ ... [FREE SOLUTION]

Chapter 6: Problem 14

For Exercises 7 through $26,$ find the area under the standard normal distribution curve. $$ \text { To the left of } z=-0.75 $$

Short Answer

Expert verified

The area to the left of $ z = -0.75 $ is approximately 0.2266.

Step by step solution

Identify the Problem

We are tasked with finding the area under the standard normal distribution curve, which is a bell curve, to the left of a given z-score, specifically for $ z = -0.75 $.

Locate the Z-score

The z-score represents the number of standard deviations a particular value is from the mean in a standard normal distribution. Here, our z-score is $ -0.75 $.

Use the Z-table

To find the area to the left of $ z = -0.75 $, we need to use the standard normal distribution table, commonly known as the Z-table. This table provides the cumulative probability from the far left to any given z-score on the distribution curve.

Read the Z-table

Locate $ -0.7 $ on the left side of the Z-table and $ 0.05 $ (the additional hundredth digit from $ -0.75 $) on the top. The table value at this intersection provides the cumulative probability, which is the area to the left.

Interpret the Table Value

The Z-table value corresponding to $ z = -0.75 $ is approximately $ 0.2266 $. This means that about 22.66% of data in a standard normal distribution falls to the left of $ z = -0.75 $.

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Key Concepts

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

z-score

The term "z-score" is vital to understanding the standard normal distribution. It's a way to describe how far a specific data point is from the mean of the distribution. Think of it as a measure of how unusual or typical a value is in a dataset.

The z-score can be calculated using the formula: \[ z = \frac{(X - \mu)}{\sigma} \]Where:
- $ X $ is the value in the dataset,
- $ \mu $ is the mean of the dataset,
- $ \sigma $ is the standard deviation of the dataset.

A positive z-score indicates a value above the mean, while a negative z-score signifies a value below the mean. In our exercise, the z-score is $-0.75$, meaning this value falls below the mean, approximately three-quarters of a standard deviation away.
Grasping the concept of z-scores will allow you to interpret where specific values land within a distribution, aiding in probability calculations and data analysis.

Z-table

The Z-table, or standard normal distribution table, is a critical tool in statistics, especially when dealing with z-scores. This table helps you find the cumulative probability associated with a particular z-score.

Each entry in the Z-table represents the area (or the cumulative probability) to the left of a given z-score on the standard normal distribution curve.
For negative z-scores, locate the corresponding row for the tenths digit (e.g., $-0.7$ from $-0.75$) and the additional hundredths digit at the top of the table (e.g., 0.05 for $-0.75$).

Finding the intersection provides the cumulative probability. In practice, the Z-table gives a quick way to determine how much of the data falls below a certain z-score without having to integrate over the distribution curve manually. It's a handy shortcut when dealing with standard normal distribution questions.

cumulative probability calculation

Cumulative probability is an essential concept that helps you quantify uncertainty. It's the probability that a random variable is less than or equal to a given value, capturing the entire distribution's probability up to a specific z-score.

For a given z-score, cumulative probability is represented as the area under the standard normal curve to the left of that z-score.
Utilizing the Z-table, you can easily find this area by noting the value that corresponds with your z-score, summarizing the percentage of data points expected to be below that score.

For instance, when finding the area to the left of $ z=-0.75 $, the cumulative probability was approximately $ 0.2266 $. This means about 22.66% of observations for a standard normal distribution fall below $ z = -0.75 $. Understanding and using cumulative probabilities enable you to solve problems involving data distribution and make informed decisions based on probabilistic outcomes.

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91影视

For Exercises 7 through \(26,\) find the area under the standard normal distribution curve. $$ \text { To the left of } z=-0.75 $$

Short Answer

Step by step solution

Identify the Problem

Locate the Z-score

Use the Z-table

Read the Z-table

Interpret the Table Value

Key Concepts

z-score

Z-table

cumulative probability calculation

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