Determine The Area Under The Standard Normal Curve That Lies Between $z=-2.09$ And $z=1.72$.---The Average Waist Size For Teenage Males Is 29 Inches With A Standard Deviation Of 2 Inches. What Is The

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Understanding the Standard Normal Curve

The standard normal curve, also known as the z-distribution, is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is used to model the distribution of many natural phenomena, including the heights of people, the weights of objects, and the times it takes for events to occur. In this article, we will determine the area under the standard normal curve that lies between z=-2.09 and z=1.72.

The Average Waist Size for Teenage Males

The average waist size for teenage males is 29 inches with a standard deviation of 2 inches. This means that the waist sizes of teenage males are normally distributed with a mean of 29 inches and a standard deviation of 2 inches. We can use the z-score formula to convert the waist sizes to z-scores:

z = (X - μ) / σ

where X is the waist size, μ is the mean, and σ is the standard deviation.

Converting Waist Sizes to Z-Scores

Let's convert the waist sizes to z-scores using the z-score formula:

z = (X - 29) / 2

For example, if the waist size is 25 inches, the z-score would be:

z = (25 - 29) / 2 = -2

If the waist size is 33 inches, the z-score would be:

z = (33 - 29) / 2 = 2

Determine the Area Under the Standard Normal Curve

To determine the area under the standard normal curve that lies between z=-2.09 and z=1.72, we can use a z-table or a calculator. A z-table is a table that shows the area under the standard normal curve to the left of a given z-score. We can use the z-table to find the area under the standard normal curve that lies between z=-2.09 and z=1.72.

Using a Z-Table

Let's use a z-table to find the area under the standard normal curve that lies between z=-2.09 and z=1.72. We can look up the z-scores in the z-table and find the corresponding areas.

For z=-2.09, the area to the left is approximately 0.0181.

For z=1.72, the area to the left is approximately 0.9584.

Finding the Area Between the Z-Scores

To find the area between the z-scores, we can subtract the area to the left of the smaller z-score from the area to the left of the larger z-score:

Area = 0.9584 - 0.0181 = 0.9403

Conclusion

In this article, we determined the area under the standard normal curve that lies between z=-2.09 and z=1.72. We used a z-table to find the area under the standard normal curve that lies between the z-scores. The area between the z-scores is approximately 0.9403.

References

  • Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Ross, S. M. (2014). Introduction to probability models. Academic Press.

Additional Resources

  • Z-table: A table that shows the area under the standard normal curve to the left of a given z-score.
  • Calculator: A calculator that can be used to find the area under the standard normal curve.
  • Online resources: There are many online resources available that can be used to find the area under the standard normal curve, including z-tables and calculators.

Frequently Asked Questions

  • Q: What is the standard normal curve? A: The standard normal curve is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1.
  • Q: How do I convert waist sizes to z-scores? A: You can use the z-score formula to convert waist sizes to z-scores: z = (X - μ) / σ.
  • Q: How do I find the area under the standard normal curve that lies between z-scores? A: You can use a z-table or a calculator to find the area under the standard normal curve that lies between z-scores.
    Frequently Asked Questions About the Standard Normal Curve =============================================================

Q: What is the standard normal curve?

A: The standard normal curve is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is used to model the distribution of many natural phenomena, including the heights of people, the weights of objects, and the times it takes for events to occur.

Q: How do I convert waist sizes to z-scores?

A: You can use the z-score formula to convert waist sizes to z-scores: z = (X - μ) / σ, where X is the waist size, μ is the mean, and σ is the standard deviation.

Q: How do I find the area under the standard normal curve that lies between z-scores?

A: You can use a z-table or a calculator to find the area under the standard normal curve that lies between z-scores. A z-table is a table that shows the area under the standard normal curve to the left of a given z-score.

Q: What is the difference between a z-table and a calculator?

A: A z-table is a table that shows the area under the standard normal curve to the left of a given z-score, while a calculator can be used to find the area under the standard normal curve that lies between z-scores.

Q: How do I use a z-table to find the area under the standard normal curve?

A: To use a z-table to find the area under the standard normal curve, you need to look up the z-score in the table and find the corresponding area. The area to the left of the z-score is listed in the table.

Q: What is the significance of the standard normal curve?

A: The standard normal curve is significant because it is used to model the distribution of many natural phenomena, including the heights of people, the weights of objects, and the times it takes for events to occur. It is also used in statistics to make inferences about populations based on samples.

Q: How do I apply the standard normal curve in real-life situations?

A: The standard normal curve can be applied in real-life situations such as:

  • Modeling the distribution of exam scores
  • Predicting the likelihood of a certain event occurring
  • Making inferences about populations based on samples

Q: What are some common applications of the standard normal curve?

A: Some common applications of the standard normal curve include:

  • Quality control
  • Reliability engineering
  • Finance
  • Medicine

Q: How do I calculate the z-score for a given value?

A: To calculate the z-score for a given value, you need to use the z-score formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

Q: What is the difference between a z-score and a standard deviation?

A: A z-score is a measure of how many standard deviations an observation is away from the mean, while a standard deviation is a measure of the spread of the data.

Q: How do I interpret the results of a z-test?

A: To interpret the results of a z-test, you need to look at the p-value and the z-score. If the p-value is less than a certain significance level (usually 0.05), you can reject the null hypothesis and conclude that the observed difference is statistically significant.

Q: What is the significance level in a z-test?

A: The significance level in a z-test is the maximum probability of rejecting the null hypothesis when it is true. It is usually set at 0.05, but can be adjusted depending on the situation.

Q: How do I choose the significance level for a z-test?

A: The significance level for a z-test should be chosen based on the research question and the level of precision desired. A smaller significance level will result in a more conservative test, while a larger significance level will result in a more liberal test.