Question

part 1 of 4
(a) find the range.
the range is 62.

part: 1 / 4

part 2 of 4
(b) find the variance and the standard deviation. assume the data represent samples, and use the shortcut formula for the unbiased estimator to compute the variance and standard deviation. round your answers to at least one decimal place.

the variance is
the standard deviation is

Explanation:

To solve for the variance and standard deviation (for a sample), we need the dataset. Since the dataset isn't provided, we'll outline the general steps using the shortcut formula for the unbiased estimator (sample variance):

Step 1: Recall the Shortcut Formula for Sample Variance

The formula for the sample variance \( s^2 \) is:
\[
s^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n - 1}
\]
where:

• \( \sum x \) = sum of all data points,
• \( \sum x^2 \) = sum of the squares of all data points,
• \( n \) = number of data points.

Step 2: Calculate the Standard Deviation

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

Example (If Dataset Were Provided)

Suppose the dataset is \( \{10, 20, 30, 40\} \) (for illustration):

• \( \sum x = 10 + 20 + 30 + 40 = 100 \)
• \( \sum x^2 = 10^2 + 20^2 + 30^2 + 40^2 = 100 + 400 + 900 + 1600 = 3000 \)
• \( n = 4 \)

Calculate Variance:

\[
s^2 = \frac{3000 - \frac{(100)^2}{4}}{4 - 1} = \frac{3000 - \frac{10000}{4}}{3} = \frac{3000 - 2500}{3} = \frac{500}{3} \approx 166.7
\]

Calculate Standard Deviation:

\[
s = \sqrt{\frac{500}{3}} \approx 12.9
\]

Next Steps (With Actual Dataset)

To solve your problem, follow these steps:

1. Identify all data points.
2. Compute \( \sum x \) (sum of data points).
3. Compute \( \sum x^2 \) (sum of squared data points).
4. Substitute \( \sum x \), \( \sum x^2 \), and \( n \) into the variance formula.
5. Take the square root of the variance to find the standard deviation.

If you provide the dataset, we can compute the exact values. For now, use the formula above with your data.

Answer:

To solve for the variance and standard deviation (for a sample), we need the dataset. Since the dataset isn't provided, we'll outline the general steps using the shortcut formula for the unbiased estimator (sample variance):

Step 1: Recall the Shortcut Formula for Sample Variance

The formula for the sample variance \( s^2 \) is:
\[
s^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n - 1}
\]
where:

• \( \sum x \) = sum of all data points,
• \( \sum x^2 \) = sum of the squares of all data points,
• \( n \) = number of data points.

Step 2: Calculate the Standard Deviation

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

Example (If Dataset Were Provided)

Suppose the dataset is \( \{10, 20, 30, 40\} \) (for illustration):

• \( \sum x = 10 + 20 + 30 + 40 = 100 \)
• \( \sum x^2 = 10^2 + 20^2 + 30^2 + 40^2 = 100 + 400 + 900 + 1600 = 3000 \)
• \( n = 4 \)

Calculate Variance:

\[
s^2 = \frac{3000 - \frac{(100)^2}{4}}{4 - 1} = \frac{3000 - \frac{10000}{4}}{3} = \frac{3000 - 2500}{3} = \frac{500}{3} \approx 166.7
\]

Calculate Standard Deviation:

\[
s = \sqrt{\frac{500}{3}} \approx 12.9
\]

Next Steps (With Actual Dataset)

To solve your problem, follow these steps:

1. Identify all data points.
2. Compute \( \sum x \) (sum of data points).
3. Compute \( \sum x^2 \) (sum of squared data points).
4. Substitute \( \sum x \), \( \sum x^2 \), and \( n \) into the variance formula.
5. Take the square root of the variance to find the standard deviation.

If you provide the dataset, we can compute the exact values. For now, use the formula above with your data.