Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Previous problemNext problem

1:33 minutes

Problem 5.1.42

Textbook Question

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the probability of z occurring in the shaded region of the standard normal distribution. The shaded region is between z = 0 and z = 1.96.

Step 2: Recall that the standard normal distribution has a mean of 0 and a standard deviation of 1. The z-scores represent the number of standard deviations away from the mean.

Step 3: Use the cumulative probability function for the standard normal distribution to find the area under the curve from z = 0 to z = 1.96. This can be done using statistical tables (z-tables) or technology such as a calculator or software.

Step 4: Look up the cumulative probability for z = 1.96 in the z-table or calculate it using technology. This value represents the area under the curve from z = -∞ to z = 1.96.

Step 5: Subtract the cumulative probability for z = 0 (which is 0.5, as it is the midpoint of the standard normal distribution) from the cumulative probability for z = 1.96 to find the probability of z occurring in the shaded region.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the z-score, which indicates how many standard deviations an element is from the mean. The area under the curve represents probabilities, with the total area equaling 1.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In the context of the standard normal distribution, z-scores allow us to determine the probability of a value falling within a certain range.

Probability and Area Under the Curve

In statistics, the probability of a z-score occurring within a certain range in the standard normal distribution is represented by the area under the curve for that range. This area can be calculated using z-tables or technology, and it reflects the likelihood of a random variable falling within that specified range.

Watch next

Master Finding Standard Normal Probabilities using z-Table with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months.

c. What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?

views

Textbook Question

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What is the highest score that would still place a person in the bottom 10% of the scores?

views

Textbook Question

Find each probability using the standard normal distribution.

b. P(z < 2.23)

views

Textbook Question

The random variable x is normally distributed with the given parameters. Find each probability.

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

views

Textbook Question

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.

views

Textbook Question

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

views

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Advanced Dental Admission Test The Advanced Dental Admission Test (ADAT) is designed so that the scores fit a normal distribution, as shown in the figure. (Source: American Dental Association)

b. Between what two values does the middle 50% of the ADAT scores lie?

views

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Red Blood Cell Count The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.4 million cells per microliter and a standard deviation of 0.4 million cells per microliter.

a. What is the minimum red blood cell count that can be in the top 25% of counts?

views

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap