Questions: The formula for the standard deviation of a sample is: s = sqrt((1/(n-1)) * sum from i=1 to n of (Xi - Xbar)^2) Select the true statement for the following data set that has a mean of 6.75: 4,6,7,10 Answer choices are rounded to the hundredths place. - The variance is 2.50 and the standard deviation is 6.50. - The variance is 6.25 and the standard deviation is 2.50. - The variance is 4.71 and the standard deviation is 2.17 - The variance is 6.75 and the standard deviation is 6.25

Solution Steps

To determine the correct statement, we need to calculate the variance and standard deviation of the given data set. The variance is the average of the squared differences from the mean, and the standard deviation is the square root of the variance.

1. Calculate the mean of the data set (already given as 6.75).
2. Compute the squared differences from the mean for each data point.
3. Find the average of these squared differences (this is the variance).
4. Take the square root of the variance to get the standard deviation.
5. Compare the calculated values with the given answer choices.

The mean of the data set is given as \( \bar{X} = 6.75 \).

For each data point \( X_i \) in the data set \([4, 6, 7, 10]\), we calculate the squared difference from the mean: \[ (X_i - \bar{X})^2 \] \[ (4 - 6.75)^2 = 7.5625 \] \[ (6 - 6.75)^2 = 0.5625 \] \[ (7 - 6.75)^2 = 0.0625 \] \[ (10 - 6.75)^2 = 10.5625 \]

The variance \( s^2 \) is the average of these squared differences, divided by \( n-1 \) where \( n \) is the number of data points: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \] \[ s^2 = \frac{1}{4-1} (7.5625 + 0.5625 + 0.0625 + 10.5625) \] \[ s^2 = \frac{1}{3} \times 18.75 = 6.25 \]

The standard deviation \( s \) is the square root of the variance: \[ s = \sqrt{s^2} = \sqrt{6.25} = 2.50 \]

Final Answer