Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.

N = 6
μ = 5
σ ≈ 2

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Step 1: Understand the problem. You are tasked with constructing a data set of size N = 6 (6 data points) such that the mean (μ) is 5 and the standard deviation (σ) is approximately 2. Recall that the mean is the average of the data points, and the standard deviation measures the spread of the data around the mean.

Step 2: Start by using the formula for the mean: μ = (Σxᵢ) / N. Rearrange this formula to find the sum of the data points: Σxᵢ = μ × N. Substitute μ = 5 and N = 6 to calculate the total sum of the data points, which will guide the construction of the data set.

Step 3: Use the formula for the standard deviation: σ = √((Σ(xᵢ - μ)²) / N). Rearrange this formula to express the sum of squared deviations: Σ(xᵢ - μ)² = σ² × N. Substitute σ ≈ 2 and N = 6 to calculate the total sum of squared deviations from the mean.

Step 4: Construct a data set of 6 numbers that satisfies both the mean and standard deviation conditions. Start with a set of numbers that sum to the total calculated in Step 2. Adjust the numbers so that their squared deviations from the mean sum to the value calculated in Step 3. Ensure the spread of the numbers is consistent with the desired standard deviation.

Step 5: Verify your constructed data set. Check that the mean of the data set is 5 by calculating (Σxᵢ) / N. Then, calculate the standard deviation using the formula σ = √((Σ(xᵢ - μ)²) / N) to ensure it is approximately 2. Adjust the data points if necessary to meet both conditions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Population Size (N)

Population size, denoted as N, refers to the total number of observations or data points in a data set. In this context, N = 6 indicates that the constructed data set must contain exactly six values. Understanding population size is crucial for calculating various statistics, as it directly influences measures like the mean and standard deviation.

Mean (μ)

The mean, represented by the symbol μ, is the average of a data set and is calculated by summing all the values and dividing by the number of observations. In this case, μ = 5 means that the average of the six data points must equal 5. This concept is fundamental in statistics as it provides a central value around which the data points are distributed.

Standard Deviation (σ)

Standard deviation, denoted as σ, measures the amount of variation or dispersion in a set of values. An approximate standard deviation of 2 indicates that the data points tend to deviate from the mean (5) by about 2 units. This concept is essential for understanding the spread of data and how individual data points relate to the mean.

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.N = 6μ = 5σ ≈ 2