A Survey Was Conducted To Measure The Number Of Hours Per Week Adults In The United States Spend on The Internet. In The Survey, The Number Of Hours Were Normally Distributed, With A Mean Of 10 Hours and A Standard Deviation Of 1.5 Hours. A Survey

by ADMIN 248 views

Introduction

In today's digital age, the internet has become an integral part of our daily lives. With the increasing use of the internet, it is essential to understand how people spend their time online. A survey was conducted to measure the number of hours per week adults in the United States spend on the internet. The survey revealed that the number of hours spent online follows a normal distribution, with a mean of 10 hours and a standard deviation of 1.5 hours. In this article, we will delve into the world of normal distribution and explore how it can be used to analyze the survey results.

What is Normal Distribution?

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In other words, the normal distribution is a bell-shaped curve where the majority of the data points are clustered around the mean, and the probability of data points decreases as you move away from the mean.

Properties of Normal Distribution

The normal distribution has several properties that make it a useful tool for data analysis. Some of the key properties of normal distribution include:

  • Symmetry: The normal distribution is symmetric about the mean, which means that the data points on one side of the mean are a mirror image of the data points on the other side.
  • Bell-shaped curve: The normal distribution has a bell-shaped curve, with the majority of the data points clustered around the mean.
  • Mean: The mean of the normal distribution is the average value of the data points.
  • Standard deviation: The standard deviation of the normal distribution is a measure of the spread of the data points from the mean.
  • Probability density function: The probability density function (PDF) of the normal distribution is a mathematical function that describes the probability of a data point occurring at a given value.

Understanding the Survey Results

The survey results show that the number of hours spent online follows a normal distribution, with a mean of 10 hours and a standard deviation of 1.5 hours. This means that the majority of the adults in the United States spend around 10 hours per week online, and the probability of spending more or less than 10 hours decreases as you move away from the mean.

Calculating the Probability of Spending More or Less Than 10 Hours

To calculate the probability of spending more or less than 10 hours, we can use the z-score formula:

z = (X - μ) / σ

where X is the value we want to calculate the probability for, μ is the mean, and σ is the standard deviation.

For example, let's say we want to calculate the probability of spending more than 12 hours online. We can plug in the values as follows:

z = (12 - 10) / 1.5 z = 2 / 1.5 z = 1.33

Using a z-table or a calculator, we can find the probability of spending more than 12 hours online. The z-score of 1.33 corresponds to a probability of approximately 0.0918. This means that about 9.18% of the adults in the United States spend more than 12 hours per week online.

Calculating the Probability of Spending Less Than 8 Hours

Similarly, we can calculate the probability of spending less than 8 hours online by plugging in the values as follows:

z = (8 - 10) / 1.5 z = -2 / 1.5 z = -1.33

Using a z-table or a calculator, we can find the probability of spending less than 8 hours online. The z-score of -1.33 corresponds to a probability of approximately 0.0918. This means that about 9.18% of the adults in the United States spend less than 8 hours per week online.

Conclusion

In conclusion, the survey results show that the number of hours spent online follows a normal distribution, with a mean of 10 hours and a standard deviation of 1.5 hours. By using the z-score formula, we can calculate the probability of spending more or less than 10 hours online. The results show that about 9.18% of the adults in the United States spend more than 12 hours per week online, and about 9.18% spend less than 8 hours per week online. This information can be useful for businesses and organizations that want to understand how people spend their time online and tailor their marketing strategies accordingly.

References

Further Reading

  • [1] "The Normal Distribution" by Michael J. Crawley
  • [2] "Statistics for Dummies" by Deborah J. Rumsey
  • [3] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole

Glossary

  • Mean: The average value of a set of data.
  • Standard deviation: A measure of the spread of a set of data from the mean.
  • Normal distribution: A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
  • Z-score: A measure of how many standard deviations an observation is away from the mean.
  • Probability density function: A mathematical function that describes the probability of a data point occurring at a given value.
    Frequently Asked Questions (FAQs) About Normal Distribution ===========================================================

Q: What is normal distribution?

A: Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Q: What are the properties of normal distribution?

A: The normal distribution has several properties that make it a useful tool for data analysis. Some of the key properties of normal distribution include:

  • Symmetry: The normal distribution is symmetric about the mean, which means that the data points on one side of the mean are a mirror image of the data points on the other side.
  • Bell-shaped curve: The normal distribution has a bell-shaped curve, with the majority of the data points clustered around the mean.
  • Mean: The mean of the normal distribution is the average value of the data points.
  • Standard deviation: The standard deviation of the normal distribution is a measure of the spread of the data points from the mean.
  • Probability density function: The probability density function (PDF) of the normal distribution is a mathematical function that describes the probability of a data point occurring at a given value.

Q: How is normal distribution used in real-life scenarios?

A: Normal distribution is used in a wide range of real-life scenarios, including:

  • Finance: Normal distribution is used to model stock prices and returns.
  • Insurance: Normal distribution is used to model claims and losses.
  • Quality control: Normal distribution is used to model defects and errors.
  • Medical research: Normal distribution is used to model patient outcomes and responses to treatment.

Q: How do I calculate the probability of a data point occurring at a given value?

A: To calculate the probability of a data point occurring at a given value, you can use the z-score formula:

z = (X - μ) / σ

where X is the value you want to calculate the probability for, μ is the mean, and σ is the standard deviation.

Q: What is the z-score?

A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:

z = (X - μ) / σ

Q: How do I interpret the z-score?

A: The z-score can be interpreted as follows:

  • Positive z-score: The observation is above the mean.
  • Negative z-score: The observation is below the mean.
  • Zero z-score: The observation is equal to the mean.

Q: What is the probability density function (PDF)?

A: The probability density function (PDF) is a mathematical function that describes the probability of a data point occurring at a given value. It is used to calculate the probability of a data point occurring at a given value.

Q: How do I calculate the probability density function (PDF)?

A: To calculate the probability density function (PDF), you can use the formula:

f(x) = (1 / σ√(2π)) * e(-(x-μ)2 / 2σ^2)

where x is the value you want to calculate the probability for, μ is the mean, and σ is the standard deviation.

Q: What is the difference between normal distribution and other types of distributions?

A: Normal distribution is different from other types of distributions in that it is symmetric about the mean, has a bell-shaped curve, and has a specific probability density function. Other types of distributions, such as the Poisson distribution and the binomial distribution, have different properties and are used to model different types of data.

Q: Can normal distribution be used to model non-normal data?

A: While normal distribution can be used to model non-normal data, it is not always the best choice. Normal distribution assumes that the data is symmetric about the mean, which may not be the case for non-normal data. In such cases, other types of distributions, such as the lognormal distribution or the gamma distribution, may be more suitable.

Q: How do I choose the right distribution for my data?

A: Choosing the right distribution for your data depends on the type of data you are working with and the research question you are trying to answer. You can use statistical tests, such as the Shapiro-Wilk test, to determine whether your data is normally distributed. If your data is not normally distributed, you can use other types of distributions, such as the Poisson distribution or the binomial distribution, to model it.