/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 The following table gives the fr... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 70

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.)

Short Answer

Expert verified
To determine mean, variance, and standard deviation, the given formulas should be applied: for the mean \(\mu = \frac{\sum_{i} m_i \cdot f_i}{\sum_{i} f_i}\), for the variance \(\sigma^2 = \frac{\sum_{i} m_i^2 \cdot f_i}{\sum_{i} f_i} - \mu^2\), and for the standard deviation \(\sigma = \sqrt{\sigma^2}\). After calculations, the final figures will be obtained.

Step by step solution

01

Calculation of Mean

First, calculate the mean (expected value) from the given frequencies and classes. This is done by multiplying each class with its frequency, adding these results together, and then dividing by the total frequency. The formula for the mean \(\mu\) is: \(\mu = \frac{\sum_{i} m_i \cdot f_i}{\sum_{i} f_i}\), where \(m\) represents the class values (number of errors) and \(f\) denotes the frequency (number of games).
02

Calculation of Variance

Next, the variance is determined. The variance gives a measure of how the data distributes itself about the mean. The formula for the variance \(\sigma^2\) is: \(\sigma^2 = \frac{\sum_{i} m_i^2 \cdot f_i}{\sum_{i} f_i} - \mu^2\), remember that \(\mu\) is the mean calculated in step 1. This formula calculates the mean of the squared deviations from the mean of the data set.
03

Calculation of Standard Deviation

Finally, the standard deviation is calculated. The standard deviation is the square root of the variance. The formula for the standard deviation \(\sigma\) is: \(\sigma = \sqrt{\sigma^2}\), where \(\sigma^2\) is the variance calculated in step 2. This step-wise process will lead you to the mean, variance, and standard deviation of the data provided in the frequency table.

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Key Concepts

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

Mean Calculation
The mean gives us the average number of errors per game made by the college baseball team over the season. Calculating the mean involves using the formula:
  1. Multiply each class value by its corresponding frequency.
  2. Add all these products together - this sum represents the total number of errors across all games.
  3. Divide this sum by the total number of games to find the average or mean.
For example, if a class value is 0 and the frequency is 11, then their product is 0\(*11 = 0\). Continue this for each class and frequency pair. Add those products together and divide by 45 (the total number of games) to find the mean, which gives us an overall sense of performance consistency.
Variance Calculation
Variance tells us how much the number of errors varies from the mean number of errors. It provides insights into the consistency of the team's performance. The formula for calculating variance is:
  • Calculate the square of each class value, then multiply by its frequency. This step finds the squared products for each frequency-class pair.
  • Sum these squared products to get a total.
  • Divide this sum by the total number of games to get the mean of the squared errors.
  • Subtract the square of the mean (computed previously) from this value.
This difference represents variance. A high variance means more inconsistency in the number of errors from game to game. If the variance is low, most games have errors around the average.
Standard Deviation Calculation
Standard deviation is a handy statistic that helps you understand the spread of your data. It is easier to interpret than variance because it is in the same units as the original data (in this case, the number of errors). Calculate it by: - Taking the square root of the variance. By doing this, the units return from squared errors (used in variance) back to regular errors. A smaller standard deviation indicates that most games had a number of errors close to the mean, while a larger standard deviation indicates a wider spread of errors. Understanding standard deviation helps coaches assess the stability of the team's defensive performance.

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Most popular questions from this chapter

One disadvantage of the standard deviation as a measure of dispersion is that it is a measure of absolute variability and not of relative variability. Sometimes we may need to compare the variability of two different data sets that have different units of measurement. The coefficient of variation is one such measure. The coefficient of variation, denoted by CV, expresses standard deviation as a percentage of the mean and is computed as follows: For population data: \(\mathrm{CV}=\frac{\sigma}{\mu} \times 100 \%\) For sample data: $$ \mathrm{CV}=\frac{s}{\bar{x}} \times 100 \% $$ The yearly salaries of all employees who work for a company have a mean of \(\$ 62,350\) and a standard deviation of \(\$ 6820\). The years of experience for the same employees have a mean of 15 years and a standard deviation of 2 years. Is the relative variation in the salaries larger or smaller than that in years of experience for these employees?

Refer to the data in Exercise 3.23, which contained the numbers of tornadoes that touched down in 12 states that had the most tornadoes during the period 1950 to 1994 . The data are reproduced here. \(\begin{array}{lllllllll}1113 & 2009 & 1374 & 1137 & 2110 & 1086 & 1166 & 1039 & 1673 & 2300\end{array}\) \(\begin{array}{ll}1139 & 5490\end{array}\) Find the range, variance, and standard deviation for these data.

Melissa's grade in her math class is determined by three 100 -point tests and a 200 -point final exam. To determine the grade for a student in this class, the instructor will add the four scores together and divide this sum by 5 to obtain a percentage. This percentage must be at least 80 for a grade of \(\mathrm{B}\). If Melissa's three test scores are 75,69 , and 87 , what is the minimum score she needs on the final exam to obtain a B grade?

How much does the typical American family spend to go away on vacation each year? Twenty-five randomly selected households reported the following vacation expenditures (rounded to the nearest hundred dollars) during the past year: \(\begin{array}{rrrrr}2500 & 500 & 800 & 0 & 100 \\ 0 & 200 & 2200 & 0 & 200 \\\ 0 & 1000 & 900 & 321,500 & 400 \\ 500 & 100 & 0 & 8200 & 900 \\ 0 & 1700 & 1100 & 600 & 3400\end{array}\) a. Using both graphical and numerical methods, organize and interpret these data. b. What measure of central tendency best answers the original question?

The heights of five starting players on a basketball team have a mean of 76 inches, a median of 78 inches, and a range of 11 inches. a. If the tallest of these five players is replaced by a substitute who is 2 inches taller, find the new mean, median, and range. b. If the tallest player is replaced by a substitute who is 4 inches shorter, which of the new values (mean, median, range) could you determine, and what would their new values be?
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