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Chapter 6: Problem 11

Find the area under the standard normal distribution curve. To the right of z = 0.37

Short Answer

Expert verified
The area to the right of z = 0.37 is approximately 0.3557.

Step by step solution

01

Understanding the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The total area under the curve is 1, which corresponds to 100% probability.
02

Identify the Z-Score

The z-score we are dealing with is 0.37. We need to find the area to the right of this z-score under the standard normal distribution curve.
03

Use the Z-Score Table

Consult a standard normal distribution table (z-table) to find the cumulative area to the left of z = 0.37. From the z-table, the area to the left of z = 0.37 is approximately 0.6443.
04

Calculate the Area to the Right

Since the total area under the curve is 1, the area to the right of z = 0.37 is calculated by subtracting the area to the left from 1: \(1 - 0.6443 = 0.3557\).

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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 measure that describes a value's position within a probability distribution. It tells us how far away a specific data point is from the mean. For the standard normal distribution, where mean is 0 and standard deviation is 1, the z-score can help determine the probability of a score occurring within a normal distribution.
  • A positive z-score indicates the data point is above the mean.
  • A negative z-score indicates it is below the mean.
  • For example, a z-score of 0.37 means the value is 0.37 standard deviations above the mean.
Recognizing z-scores can be especially useful when comparing scores from different normal distributions or when you need to understand the extremity of a score in a data set.
Exploring Cumulative Probability
Cumulative probability refers to the probability that a random variable will be less than or equal to a specific value. This is an essential concept in understanding how likely it is that a certain event will occur within a normal distribution. When considering the area under the standard normal distribution curve, cumulative probability is expressed as a value between 0 and 1.
  • This area represents the likelihood of a random variable falling below a particular z-score.
  • A cumulative probability close to 0 indicates a rare event, while one near 1 suggests a very likely event.
  • For instance, the cumulative probability to the left of a z-score of 0.37 is 0.6443, signifying that there is a 64.43% chance of a score falling below this point in the distribution.
Navigating the Z-Table
The z-table, also known as the standard normal distribution table, is a valuable tool for finding cumulative probabilities associated with specific z-scores. It displays the cumulative probability from the mean of the standard normal distribution up to the z-score. To use the z-table, follow these steps:
  • Locate the row that corresponds to the first decimal of your z-score (e.g., for 0.37, look for 0.3).
  • Within that row, find the column corresponding to the second decimal of your z-score (e.g., the second decimal is 0.07 for 0.37).
  • The entry in the table is the cumulative probability to the left of the z-score.
For z = 0.37, we find that the cumulative area is approximately 0.6443, meaning there is a 64.43% probability that a score is less than or equal to this z-score.
The Basics of Probability Distribution
Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. In a normal distribution, known as the bell curve:
  • The mean serves as the peak of the curve.
  • Most data points cluster around the mean, with probabilities tapering off symmetrically towards the edges.
  • The standard deviation determines the width of the curve.
In the case of the standard normal distribution:
  • The mean is 0, and the standard deviation is 1.
  • The total area under the curve is 1, representing 100% of possible probabilities.
By understanding a probability distribution, you can interpret how data is spread out and calculate the likelihood of values occurring within different sections of the curve.

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Most popular questions from this chapter

A worker in the automobile industry works an average of 43.7 hours per week. If the distribution is approximately normal with a standard deviation of 1.6 hours, what is the probability that a randomly selected automobile worker works less than 40 hours per week?

The mayor of a small town estimates that 35% of the residents in the town favor the construction of a municipal parking lot. If there are 350 people at a town meeting, find the probability that at least 100 favor construction of the parking lot. Based on your answer, is it likely that 100 or more people would favor the parking lot?

The average salary of a male full professor at a public four-year institution offering classes at the doctoral level is \(\$ 99,685 .\) For a female full professor at the same kind of institution, the salary is \(\$ 90,330\). If the standard deviation for the salaries of both genders is approximately \(\$ 5200\) and the salaries are normally distributed, find the 80th percentile salary for male professors and for female professors.

A contractor decided to build homes that will include the middle 80% of the market. If the average size of homes built is 1810 square feet, find the maximum and minimum sizes of the homes the contractor should build. Assume that the standard deviation is 92 square feet and the variable is normally distributed.

If samples of a specific size are selected from a population and the means are computed, what is this distribution of means called?
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