Questions: The following data are the final exam scores of the 14 students in a small calculus class. 84, 90, 80, 96, 89, 63, 72, 93, 71, 94, 74, 74, 70, 98 Are these population or sample data? Population What is the range for this data set? What is the variance for this data set? Round your answer to three decimal places, if necessary. What is the standard deviation for this data set? Round your answer to three decimal places, if necessary.

Solution Steps

To solve the given questions, we need to perform the following steps:

1. Range: Calculate the range by finding the difference between the maximum and minimum values in the data set.
2. Variance: Calculate the variance using the formula for population variance, which involves finding the mean of the data set, then computing the average of the squared differences from the mean.
3. Standard Deviation: Calculate the standard deviation by taking the square root of the variance.

The range of a data set is the difference between the maximum and minimum values. Given the data set: \[ \{84, 90, 80, 96, 89, 63, 72, 93, 71, 94, 74, 74, 70, 98\} \]

The maximum value is \(98\) and the minimum value is \(63\). Therefore, the range is: \[ \text{Range} = 98 - 63 = 35 \]

The variance for a population data set is calculated using the formula: \[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \] where \(N\) is the number of data points, \(x_i\) are the data points, and \(\mu\) is the mean of the data set.

First, calculate the mean (\(\mu\)): \[ \mu = \frac{1}{14} \sum_{i=1}^{14} x_i = 82.0 \]

Next, calculate the variance: \[ \sigma^2 = \frac{1}{14} \sum_{i=1}^{14} (x_i - 82.0)^2 = 120.8571 \]

The standard deviation is the square root of the variance: \[ \sigma = \sqrt{120.8571} = 10.9935 \]

Final Answer