Question

Set \(X\) consists of at least 2 members and is a set of consecutive odd integers with an average (arithmetic mean) of 37. Set \(Y\) consists of at least 10 members and is also a set of consecutive odd integers with an average (arithmetic mean) of 37. Set \(Z\) consists of all of the members of both set \(X\) and set \(Y\). Which of the following statements must be true? I. The standard deviation of set \(Z\) is not equal to the standard deviation of set \(X\). II. The standard deviation of set \(Z\) is equal to the standard deviation of set \(Y\). III. The average (arithmetic mean) of set \(Z\) is 37. a. I only b. II only c. III only d. I and III e. II and III

Short answer

The correct answer is d. I and III

Step-by-step solution

Understand Set Structure

Set X and Set Y are both sets of consecutive odd integers with the same average of 37. Set X has at least 2 members, and Set Y has at least 10 members.

Calculation of Arithmetic Mean

Both sets X and Y have the arithmetic mean of 37. This implies the sum of the elements in each set divided by the number of elements equals 37.

Combination into Set Z

Set Z consists of all members of both Set X and Set Y. Hence, Set Z contains the combined elements of X and Y.

Mean of Set Z

Since the arithmetic mean of both sets is the same, the arithmetic mean of Set Z, which is formed by elements of Set X and Set Y, remains 37.

Evaluate Standard Deviation

The standard deviation depends not only on the values but also on the number of elements in the set. Since Set Y has more elements compared to Set X, the standard deviation of Z will not be equal to the standard deviation of X.

Equal Standard Deviation Analysis

Because Set Y has a standard deviation based on more elements and includes all of X's values, the spread of Z should be similar to Y but different from X due to the number of elements. Thus, Standard deviation of Z equal to X is false.

Final Solution

Analyzing the provided statements: I is correct since the standard deviations do not match due to different sizes. II is incorrect as explained. III is correct as the combined mean is 37.

Key concepts

Consecutive Odd Integers

Consecutive odd integers are numbers that follow one another in order and are all odd.
Examples include 1, 3, 5, 7, and so forth. Each number in this series is two units apart from the previous number.
In the given problem, both sets \(X\) and \(Y\) consist of consecutive odd integers.
This means that within each set, each number is separated by 2.

For set theory and solution solving, knowing that these integers are consecutive helps to simplify many calculations.

Set Theory

Sets are collections of distinct objects. In this exercise, we deal with three sets: \(X\), \(Y\), and \(Z\).
Set \(X\) includes at least 2 consecutive odd integers with an arithmetic mean of 37.
Set \(Y\) includes at least 10 consecutive odd integers with the same mean.
Set \(Z\) is formed by combining elements from both \(X\) and \(Y\).

Understanding set theory helps in analyzing the characteristics of combined sets.
It also assists in understanding how individual set properties affect the combined set \(Z\).

Statistical Analysis

Statistical analysis involves collecting and analyzing data to understand patterns and trends.
For this problem, the focus is on determining different measures like the mean and standard deviation for the sets.
The arithmetic mean indicates the central point of the data.
The standard deviation assesses the spread or variability around the mean.

By combining and analyzing the given sets, statistical properties can be better understood.
This helps in answering questions about the similarities and differences among the sets.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values.
In mathematical terms, it’s expressed as:
\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2} \]
Here, \( \sigma\) represents the standard deviation, \(N\) is the number of elements, \(x_i\) are the individual elements, and \( \mu\) is the mean of the elements.

In the given problem, set \(X\) has fewer elements than set \(Y\), thus changing the spread.
Combining both sets into \(Z\) results in a standard deviation that's distinct from both \(X\) and more similar to \(Y\) because \(Y\) has more members.

Mean Calculation

The arithmetic mean is a statistical measure to calculate the average of a set of numbers.
The formula is given by:
\[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \]
Here, \( \mu\) represents the mean, \(N\) is the number of values, and \(x_i\) are individual elements.

In the exercise, both sets \(X\) and \(Y\) have their means as 37.
To find the mean of set \(Z\), you combine elements from both sets.
Since they already have the same mean, the overall mean remains 37.
This is a key property of arithmetic mean when combining sets with identical mean values.