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

1:42 minutes

Problem 5.RE.2

Textbook Question

In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.

Verified step by step guidance

Step 1: Observe the normal curve provided in the image. The curve is symmetric, which indicates that the data follows a normal distribution. The peak of the curve represents the mean (μ), and the spread of the curve is determined by the standard deviation (σ).

Step 2: Identify the mean (μ) from the graph. The mean is located at the center of the curve, which corresponds to the value of 55 on the x-axis.

Step 3: Estimate the standard deviation (σ) using the red dashed lines. These lines mark one standard deviation away from the mean on both sides. From the graph, the values at one standard deviation are approximately 50 and 60.

Step 4: Calculate the standard deviation (σ) by finding the distance between the mean and one of the dashed lines. For example, σ = |55 - 50| = 5 or σ = |60 - 55| = 5.

Step 5: Summarize the findings: The mean (μ) is 55, and the standard deviation (σ) is 5. These values describe the normal distribution represented by the curve.

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.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three, making it essential for statistical analysis.

Mean

The mean is the average value of a set of numbers, calculated by summing all values and dividing by the count of values. In the context of a normal distribution, the mean represents the center of the distribution, where the highest point of the curve occurs. It is a measure of central tendency that provides a useful summary of the data set.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. In the context of the normal distribution, it helps determine the width of the curve and the proportion of data within specific ranges around the mean.

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

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

Between z = -1.55 and z = 1.04

views

Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = -2.825

views

Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = -1.95

views

Textbook Question

In Exercises 5 and 6, find the area of the indicated region under the standard normal curve. If convenient, use technology to find the area.

views

Textbook Question

Employee Wellness A survey of employed U.S. adults found that only 35% believe their employer cares about their well-being. You randomly select a sample of U.S. employees. Find the probability that fewer than 100 believe their employer cares about their well-being. (Source: Gallup)

c. You select 400 U.S. employees.

views

Textbook Question

Daily Commute About 83% of U.S. employees drive their own vehicle to work. You randomly select a sample of U.S. employees. Find the probability that more than 100 of the employees drive their own vehicle to work. (Source: U.S. Bureau of Labor Statistics)

c. You select 150 U.S. employees.

views

Textbook Question

Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.

views

Textbook Question

Determine whether any of the events in Exercise 33 are unusual. Explain your reasoning.

views

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap