Question

a local inn asked customers to rate their stays on a scale of 1-10. the following list shows the ratings given by the last 10 customers.

4, 3, 1, 8, 9, 5, 5, 8, 3, 5

complete parts a) through c).

a) calculate the range.
the range is

b) calculate the variance.
the sample variance is
(round to two decimal places as needed.)

c) calculate the standard deviation.
the sample standard deviation is
(round to two decimal places as needed.)

Explanation:

⚡ Using: Variance_Definition · 🆕 New: Range and Standard Deviation
How spread out data values are

Step 1: Calculate the Range

The data set of the last 10 customer ratings is:

$$ \{4, 3, 1, 8, 9, 5, 5, 8, 3, 5\} $$

To find the range, subtract the minimum value from the maximum value:

• Maximum value = \(9\)
• Minimum value = \(1\)

$$ \text{Range} = \text{Maximum} - \text{Minimum} $$

$$ \text{Range} = 9 - 1 = 8 $$

Step 2: Calculate the Sample Mean

To calculate the sample variance, we first need the sample mean (\(\bar{x}\)):

$$ \bar{x} = \frac{\sum x}{n} $$

$$ \bar{x} = \frac{4 + 3 + 1 + 8 + 9 + 5 + 5 + 8 + 3 + 5}{10} $$

$$ \bar{x} = \frac{50}{10} = 5 $$

Step 3: Calculate the Sample Variance

The formula for sample variance (\(s^2\)) is:

$$ s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} $$

Let's find the squared deviations from the mean (\(\bar{x} = 5\)) for each data point:

• \((4 - 5)^2 = (-1)^2 = 1\)
• \((3 - 5)^2 = (-2)^2 = 4\)
• \((1 - 5)^2 = (-4)^2 = 16\)
• \((8 - 5)^2 = (3)^2 = 9\)
• \((9 - 5)^2 = (4)^2 = 16\)
• \((5 - 5)^2 = (0)^2 = 0\)
• \((5 - 5)^2 = (0)^2 = 0\)
• \((8 - 5)^2 = (3)^2 = 9\)
• \((3 - 5)^2 = (-2)^2 = 4\)
• \((5 - 5)^2 = (0)^2 = 0\)

Sum of squared deviations:

$$ \sum (x - \bar{x})^2 = 1 + 4 + 16 + 9 + 16 + 0 + 0 + 9 + 4 + 0 = 59 $$

Divide by \(n - 1\) (where \(n = 10\)):

$$ s^2 = \frac{59}{10 - 1} = \frac{59}{9} \approx 6.5556 $$

Rounded to two decimal places, the sample variance is \(6.56\).

Step 4: Calculate the Sample Standard Deviation

The sample standard deviation (\(s\)) is the square root of the sample variance:

$$ s = \sqrt{s^2} $$

$$ s = \sqrt{\frac{59}{9}} \approx 2.5604 $$

Rounded to two decimal places, the sample standard deviation is \(2.56\).

Answer:

a) The range is 8

b) The sample variance is 6.56

c) The sample standard deviation is 2.56