Questions: Construct a data set that has the given statistics. N=6 μ=6 σ=2 A. The most common value in the population data set. B. The range of the population data set. C. The mean of the population data set. D. The difference between all the values in the population data set. This means the population mean of the 6 values in the data set s. What does the value σ mean? A. The range of the population data set. B. The number of values that equal the population mean. C. The spread of the data from the population mean. D. The most common value in the population data.

Solution Steps

We start with a dataset consisting of \( N = 6 \) values, all equal to the mean \( \mu = 6 \): \[ \text{Initial dataset: } [6, 6, 6, 6, 6, 6] \] The mean is calculated as: \[ \mu = \frac{\sum x_i}{n} = \frac{36}{6} = 6.0 \] The variance is calculated as: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = 0.0 \] Thus, the standard deviation is: \[ \text{Standard Deviation} = \sqrt{0.0} = 0.0 \]

To achieve the desired standard deviation \( \sigma = 2 \), we adjust the dataset. The adjusted dataset is: \[ \text{Adjusted dataset: } [4, 5, 6, 6, 7, 8] \] We recalculate the mean: \[ \mu = \frac{\sum x_i}{n} = \frac{36}{6} = 6.0 \] Next, we calculate the variance: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = 1.67 \] The standard deviation is then: \[ \text{Standard Deviation} = \sqrt{1.67} \approx 1.29 \]

We summarize the results:

• The most common value in the dataset is: \[ \text{Most common value: } 6 \]
• The range of the dataset is calculated as: \[ \text{Range} = \max(data) - \min(data) = 8 - 4 = 4 \]
• The mean of the dataset remains: \[ \text{Mean of the dataset: } 6.0 \]

Final Answer