An Insurance Company Crashed Four Cars In Succession At 5 Miles Per Hour. The Cost Of Repair For Each Of The Four Crashes Was $\$417, \$459, \$415, \$219$. Compute The Range, Sample Variance, And Sample Standard Deviation

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Introduction

In this article, we will be discussing the computation of the range, sample variance, and sample standard deviation for a given set of data. The data provided represents the cost of repair for four cars that were crashed in succession at a speed of 5 miles per hour. The costs are $$417, $459, $415, $219219. We will use these values to calculate the range, sample variance, and sample standard deviation.

Calculating the Range

The range is the difference between the largest and smallest values in a dataset. To calculate the range, we need to identify the largest and smallest values in the dataset.

  • The largest value in the dataset is $$459.
  • The smallest value in the dataset is $$219.

Now, we can calculate the range by subtracting the smallest value from the largest value.

Range = Largest value - Smallest value Range = $$459 - $$219 Range = $$240

Calculating the Sample Variance

The sample variance is a measure of the spread or dispersion of a dataset. It is calculated by finding the average of the squared differences between each value and the mean of the dataset.

First, we need to calculate the mean of the dataset.

Mean = (Sum of all values) / (Number of values) Mean = ($$417 + $$459 + $$415 + $$219) / 4 Mean = $$1510 / 4 Mean = $$377.50

Next, we need to calculate the squared differences between each value and the mean.

Squared differences:

  • ($$417 - $377.50)2=(377.50)^2 = ($39.50)^2 = $$1550.25
  • ($$459 - $377.50)2=(377.50)^2 = ($81.50)^2 = $$6642.25
  • ($$415 - $377.50)2=(377.50)^2 = ($37.50)^2 = $$1406.25
  • ($$219 - $377.50)2=(377.50)^2 = ($-158.50)^2 = $$25100.25

Now, we can calculate the sample variance by finding the average of the squared differences.

Sample variance = (Sum of squared differences) / (Number of values - 1) Sample variance = ($$1550.25 + $$6642.25 + $$1406.25 + $$25100.25) / 3 Sample variance = $$38499 / 3 Sample variance = $$12833

Calculating the Sample Standard Deviation

The sample standard deviation is the square root of the sample variance. It is a measure of the spread or dispersion of a dataset.

Sample standard deviation = √(Sample variance) Sample standard deviation = √($$12833) Sample standard deviation = $$113.04

Conclusion

In this article, we calculated the range, sample variance, and sample standard deviation for a given set of data. The data represented the cost of repair for four cars that were crashed in succession at a speed of 5 miles per hour. The costs were $$417, $$459, $$415, $$219. We found that the range was $$240, the sample variance was $$12833, and the sample standard deviation was $$113.04.

References

Introduction

In this article, we will be discussing the computation of the range, sample variance, and sample standard deviation for a given set of data. The data provided represents the cost of repair for four cars that were crashed in succession at a speed of 5 miles per hour. The costs are $$417, $$459, $$415, $219219. We will use these values to calculate the range, sample variance, and sample standard deviation.

Calculating the Range

The range is the difference between the largest and smallest values in a dataset. To calculate the range, we need to identify the largest and smallest values in the dataset.

  • The largest value in the dataset is $$459.
  • The smallest value in the dataset is $$219.

Now, we can calculate the range by subtracting the smallest value from the largest value.

Range = Largest value - Smallest value Range = $$459 - $$219 Range = $$240

Calculating the Sample Variance

The sample variance is a measure of the spread or dispersion of a dataset. It is calculated by finding the average of the squared differences between each value and the mean of the dataset.

First, we need to calculate the mean of the dataset.

Mean = (Sum of all values) / (Number of values) Mean = ($$417 + $$459 + $$415 + $$219) / 4 Mean = $$1510 / 4 Mean = $$377.50

Next, we need to calculate the squared differences between each value and the mean.

Squared differences:

  • ($$417 - $377.50)2=(377.50)^2 = ($39.50)^2 = $$1550.25
  • ($$459 - $377.50)2=(377.50)^2 = ($81.50)^2 = $$6642.25
  • ($$415 - $377.50)2=(377.50)^2 = ($37.50)^2 = $$1406.25
  • ($$219 - $377.50)2=(377.50)^2 = ($-158.50)^2 = $$25100.25

Now, we can calculate the sample variance by finding the average of the squared differences.

Sample variance = (Sum of squared differences) / (Number of values - 1) Sample variance = ($$1550.25 + $$6642.25 + $$1406.25 + $$25100.25) / 3 Sample variance = $$38499 / 3 Sample variance = $$12833

Calculating the Sample Standard Deviation

The sample standard deviation is the square root of the sample variance. It is a measure of the spread or dispersion of a dataset.

Sample standard deviation = √(Sample variance) Sample standard deviation = √($$12833) Sample standard deviation = $$113.04

Conclusion

In this article, we calculated the range, sample variance, and sample standard deviation for a given set of data. The data represented the cost of repair for four cars that were crashed in succession at a speed of 5 miles per hour. The costs were $$417, $$459, $$415, $$219. We found that the range was $$240, the sample variance was $$12833, and the sample standard deviation was $$113.04.

Q&A

Q: What is the range of the dataset?

A: The range of the dataset is $240,whichisthedifferencebetweenthelargestvalue(240, which is the difference between the largest value ($459)andthesmallestvalue(459) and the smallest value ($219).

Q: How is the sample variance calculated?

A: The sample variance is calculated by finding the average of the squared differences between each value and the mean of the dataset.

Q: What is the sample standard deviation?

A: The sample standard deviation is the square root of the sample variance. It is a measure of the spread or dispersion of a dataset.

Q: What is the difference between the sample variance and the sample standard deviation?

A: The sample variance is a measure of the spread or dispersion of a dataset, while the sample standard deviation is the square root of the sample variance.

Q: Can you explain the concept of sample variance in simpler terms?

A: The sample variance is a measure of how spread out the values in a dataset are. It's like measuring how far apart the values are from the average value.

Q: How is the sample standard deviation used in real-world applications?

A: The sample standard deviation is used in a variety of real-world applications, such as finance, engineering, and quality control. It's used to measure the spread or dispersion of a dataset and to make predictions about future values.

Q: Can you provide an example of how the sample standard deviation is used in finance?

A: Yes, the sample standard deviation is used in finance to measure the risk of a stock or investment. It's used to calculate the volatility of a stock and to make predictions about future stock prices.

Q: What is the importance of understanding the sample standard deviation?

A: Understanding the sample standard deviation is important because it allows you to make informed decisions about a dataset. It helps you to understand the spread or dispersion of the data and to make predictions about future values.

References