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Chapter 3: Problem 48

The following data give the number of highway collisions with large wild animals, such as deer or moose, in one of the northeastern states during each week of a 9-week period. 7 \(\begin{array}{llllllll}10 & 3 & 8 & 2 & 5 & 7 & 4 & 9\end{array}\) Find the range, variance, and standard deviation.

Short Answer

Expert verified
The range, variance and standard deviation of the given data set are calculated corresponding to the steps above. They would be different depending on the dataset.

Step by step solution

01

Calculate the Range

The range is the difference between the greatest and smallest numbers in a data set. In this case, the numbers are: 7, 10, 3, 8, 2, 5, 7, 4, and 9. Find the highest and lowest numbers, then subtract the lowest from the highest to obtain the range.
02

Calculate the Variance

First, compute the mean (average) of the data set by adding all the numbers and then dividing by how many numbers there are. To calculate the variance, subtract the mean from each data point, square the result for each, add those values together, and then divide by the number of data points. The formula is as follows: \[Var = \frac{{\sum (x_i - \bar{x})^2}}{N}\] where \(x_i\) are the data points, \(\bar{x}\) is the mean, and N is the number of data points.
03

Calculate the Standard Deviation

The standard deviation is the square root of the variance. Hence, take the square root of the variance computed in the previous step to find the standard deviation. The formula is as follows: \[SD = \sqrt{Var}\] where Var is the variance already calculated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range
The range is the simplest way to measure the spread of a data set and gives you an idea of how far apart the smallest and largest numbers are.
  • To calculate the range, identify the largest value and the smallest value in the data set.
  • In the given data set: 10 is the largest number, and 2 is the smallest number.
  • The range is then calculated by subtracting the smallest number from the largest: \( 10 - 2 = 8 \).
The range is a useful measure because it gives a snapshot of the variability in the data, but it does not take into account how the other values are distributed. It's a good starting point and is always quick to compute.
Variance
Variance provides a measure of how much the values in a data set differ from the mean (average) of the data set. It tells us about the data's spread and is useful for understanding the distribution of data points.
  • First, calculate the mean by adding all the values and dividing by the number of values: \[ \text{Mean} = \frac{7 + 10 + 3 + 8 + 2 + 5 + 7 + 4 + 9}{9} = 6.11 \text{ (approx)} \]
  • Subtract the mean from each number to find the deviation of each number from the mean.
  • Square each deviation to eliminate negative values and add them all up.
  • Finally, divide the sum of these squared deviations by the number of data points to get the variance:
    • \[ \text{Variance} = \frac{(7-6.11)^2 + (10-6.11)^2 + \ldots + (9-6.11)^2}{9} \]
Variance helps us understand whether the data points are very far from the mean or clustered closely around it.
Standard Deviation
The standard deviation is a cornerstone concept in statistics that gives us insight into the typical distance of data points from the mean.
  • It is derived by taking the square root of the variance, which makes it more interpretable as it is in the same units as the data points.
  • Using the variance we calculated earlier, the standard deviation is:
    • \[ \text{SD} = \sqrt{\text{Variance}} \]
The standard deviation provides a clearer, more comprehensible measure of spread because compared to variance, it directly relates to the data's original scale. If the standard deviation is low, most data points are close to the mean. A high standard deviation indicates that the data points are spread out over a wider range of values.

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Most popular questions from this chapter

The annual earnings of all employees with CPA certification and 6 years of experience and working for large firms have a bell-shaped distribution with a mean of \(\$ 134,000\) and a standard deviation of \(\$ 12,000 .\) a. Using the empirical rule, find the percentage of all such employees whose annual eamings are between i. \(\$ 98,000\) and \(\$ 170,000\) ii. \(\$ 110,000\) and \(\$ 158,000\) *b. Using the empirical rule, find the interval that contains the annual earnings of \(68 \%\) of all such employees.

Briefly explain the difference between a population parameter and a sample statistic. Give one example of each.

The following table gives the frequency distribution of the number of errors committed by a college baseball team in all of the 45 games that it played during the \(2011-12\) season. \begin{tabular}{cc} \hline Number of Errors & Number of Games \\ \hline 0 & 11 \\ 1 & 14 \\ 2 & 9 \\ 3 & 7 \\ 4 & 3 \\ 5 & 1 \\ \hline \end{tabular} Find the mean, variance, and standard deviation. (Hint: The classes in this example are single valued. These values of classes will be used as values of \(m\) in the formulas for the mean, variance, and standard deviation.)

Refer to Exercise 3.115. Suppose the times taken to learn the basics of this software program by all students have a bell-shaped distribution with a mean of 200 minutes and a standard deviation of 20 minutes. a. Using the empirical rule, find the percentage of students who will learn the basics of this software program in i. 180 to 220 minutes ii. 160 to 240 minutes *b. Using the empirical rule, find the interval that contains the times taken by \(99.7 \%\) of all students to learn this software program.

The following data give the lengths of time (in weeks) taken to find a full- time job by 18 computer science majors who graduated in 2011 from a small college. \(\begin{array}{rrrrrrrrr}30 & 43 & 32 & 21 & 65 & 8 & 4 & 18 & 16 \\ 38 & 9 & 44 & 33 & 23 & 24 & 81 & 42 & 55\end{array}\) Make a box-and-whisker plot. Comment on the skewness of this data set. Does this data set contain any outliers?
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