Problem 64
Question
For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is [2010] (a) \(\frac{11}{2}\) (b) 6 (c) \(\frac{13}{2}\) (d) \(\frac{5}{2}\)
Step-by-Step Solution
Verified Answer
The combined variance is \(\frac{11}{2}\), so the answer is (a) \(\frac{11}{2}\).
1Step 1: Understand Combined Variance Formula
To find the variance of the combined data set, we use the formula for the combined variance of two groups with different variances: \(Var(Z) = \frac{n_1(Var(X) + (\bar{X} - \bar{Z})^2) + n_2(Var(Y) + (\bar{Y} - \bar{Z})^2)}{n_1 + n_2} \\)where \(\bar{Z}\) is the combined mean of the datasets, \(\bar{X}\) and \(\bar{Y}\) are the means, \(Var(X)\) and \(Var(Y)\) are the variances of the sets of size \(n_1 = 5\) and \(n_2 = 5\).
2Step 2: Calculate Combined Mean
Calculate the combined mean \(\bar{Z}\) using the formula: \(\bar{Z} = \frac{n_1 \bar{X} + n_2 \bar{Y}}{n_1 + n_2}\). Substituting the given values: \(\bar{Z} = \frac{5 \times 2 + 5 \times 4}{5 + 5} = \frac{10 + 20}{10} = 3\).
3Step 3: Individual Mean Deviations
Calculate \((\bar{X} - \bar{Z})^2\) and \((\bar{Y} - \bar{Z})^2\). For \(\bar{X} = 2\) and \(\bar{Z} = 3\), \((2 - 3)^2 = 1\). For \(\bar{Y} = 4\) and \(\bar{Z} = 3\), \((4 - 3)^2 = 1\).
4Step 4: Apply Values to Combined Variance Formula
Substitute the values into the combined variance formula: \[Var(Z) = \frac{5 (4 + 1) + 5 (5 + 1)}{10} = \frac{5 \times 5 + 5 \times 6}{10} = \frac{25 + 30}{10} = \frac{55}{10} = \frac{11}{2}\]
5Step 5: Determine the Answer
Identify the correct option based on the calculated value of the combined variance. The value is \(\frac{11}{2}\), thus the answer is option (a) \(\frac{11}{2}\).
Key Concepts
VarianceMeanMathematics Problem SolvingData Sets
Variance
Variance is a key concept in statistics that indicates how much the values in a data set differ from the mean. Essentially, it measures the spread of a set of numbers. To calculate variance, you take each number in the data set and subtract the mean, then square the result to get rid of negative differences. These squared differences are averaged to provide the variance.
In more mathematical terms, for a data set of size \( n \) with elements \( x_1, x_2, ..., x_n \), and mean \( \bar{x} \), the variance \( Var(X) \) is calculated as:
\[ Var(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]This computation illustrates how far numbers are distributed from the mean.
Varieties of variance computations exist, such as for larger populations or subgroups, but the overall idea remains consistent: determining the degree of dispersion among data points.
In more mathematical terms, for a data set of size \( n \) with elements \( x_1, x_2, ..., x_n \), and mean \( \bar{x} \), the variance \( Var(X) \) is calculated as:
\[ Var(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]This computation illustrates how far numbers are distributed from the mean.
Varieties of variance computations exist, such as for larger populations or subgroups, but the overall idea remains consistent: determining the degree of dispersion among data points.
Mean
The mean, often referred to as the average, is a fundamental measure of central tendency in mathematics. It provides a summary of the data set by representing the total sum of all values divided by the number of elements.
To find the mean of a data set, you sum all the numbers and divide by the count of numbers. For a set \( \{x_1, x_2, ..., x_n\} \), the mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]
This calculation is straightforward and helps simplify analysis by providing a common point around which the data values cluster.
The mean is particularly useful when looking out for trends or patterns, as it gives a quick overview of the data’s balance.
To find the mean of a data set, you sum all the numbers and divide by the count of numbers. For a set \( \{x_1, x_2, ..., x_n\} \), the mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]
This calculation is straightforward and helps simplify analysis by providing a common point around which the data values cluster.
The mean is particularly useful when looking out for trends or patterns, as it gives a quick overview of the data’s balance.
Mathematics Problem Solving
Mathematics problem solving involves a series of logical steps to arrive at a desired solution. When dealing with variances and means, problem solving often requires understanding various formulas and how to apply them correctly.
For instance, finding the combined variance involves numerous steps:
Problem solving in math is similar to piecing together a puzzle, where each piece represents a part of the solution process.
For instance, finding the combined variance involves numerous steps:
- Calculate individual variances and means.
- Determine the combined mean.
- Identify deviations from the combined mean.
- Apply these values in the combined variance formula.
- Simplify and solve the equation to find the answer.
Problem solving in math is similar to piecing together a puzzle, where each piece represents a part of the solution process.
Data Sets
In statistical analysis, data sets consist of various data points collected for examination and inference. A data set can include numbers, names, or any measurable elements. Each item in a data set is regarded as a data point.
Evaluating data sets involves calculating various statistical measures such as mean and variance, which help provide insights into trends, central tendencies, and variability.
Data sets can vary widely in size and type, and they often necessitate specific approaches for analysis based on the context and nature of the collected data. Understanding how to manipulate and analyze data sets is crucial to interpreting any findings accurately.
Particularly, when dealing with multiple data sets, as in the original exercise, integration through combined measures enables a more holistic understanding of the overall data behavior.
Evaluating data sets involves calculating various statistical measures such as mean and variance, which help provide insights into trends, central tendencies, and variability.
Data sets can vary widely in size and type, and they often necessitate specific approaches for analysis based on the context and nature of the collected data. Understanding how to manipulate and analyze data sets is crucial to interpreting any findings accurately.
Particularly, when dealing with multiple data sets, as in the original exercise, integration through combined measures enables a more holistic understanding of the overall data behavior.
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