Questions: The ages (in years) of a random sample of shoppers at a gaming store are shown. Determine the range, mean, variance, and standard deviation of the set. 12,18,23,13,19,17,19,16,15,19 The range is 11. (Simplify your answer.) The mean is 17.1. (Simplify your answer. Round to the nearest tenth as needed.) The variance is (Simplify your answer. Round to the nearest tenth as needed.)

The ages (in years) of a random sample of shoppers at a gaming store are shown. Determine the range, mean, variance, and standard deviation of the set.
12,18,23,13,19,17,19,16,15,19

The range is 11.
(Simplify your answer.)
The mean is 17.1.
(Simplify your answer. Round to the nearest tenth as needed.)
The variance is 
(Simplify your answer. Round to the nearest tenth as needed.)
Transcript text: The ages (in years) of a random sample of shoppers at a gaming store are shown. Determine the range, mean, variance, and standard deviation of the sar set. \[ 12,18,23,13,19,17,19,16,15,19 \] The range is 11 . (Simplify your answer.) The mean is 17.1. (Simplify your answer. Round to the nearest tenth as needed.) The variance is $\square$ (Simplify your answer. Round to the nearest tenth as needed.)
failed

Solution

failed
failed

Solution Steps

Step 1: Calculate the Range

The range of a dataset is calculated as the difference between the maximum and minimum values. For the given ages:

\[ \text{Range} = \max(x) - \min(x) = 23 - 12 = 11 \]

Step 2: Calculate the Mean

The mean (average) is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

For the given ages:

\[ \mu = \frac{12 + 18 + 23 + 13 + 19 + 17 + 19 + 16 + 15 + 19}{10} = \frac{171}{10} = 17.1 \]

Step 3: Calculate the Variance

The variance is calculated using the formula for sample variance:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

Calculating the squared differences from the mean:

\[ \begin{align_} (12 - 17.1)^2 & = 26.01 \\ (18 - 17.1)^2 & = 0.81 \\ (23 - 17.1)^2 & = 34.81 \\ (13 - 17.1)^2 & = 16.81 \\ (19 - 17.1)^2 & = 3.61 \\ (17 - 17.1)^2 & = 0.01 \\ (19 - 17.1)^2 & = 3.61 \\ (16 - 17.1)^2 & = 1.21 \\ (15 - 17.1)^2 & = 4.41 \\ (19 - 17.1)^2 & = 3.61 \\ \end{align_} \]

Summing these squared differences:

\[ \sum (x_i - \mu)^2 = 26.01 + 0.81 + 34.81 + 16.81 + 3.61 + 0.01 + 3.61 + 1.21 + 4.41 + 3.61 = 94.5 \]

Now, substituting into the variance formula:

\[ \sigma^2 = \frac{94.5}{10 - 1} = \frac{94.5}{9} = 10.5 \]

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sqrt{\sigma^2} = \sqrt{10.5} \approx 3.2 \]

Final Answer

The range is \(\boxed{11}\).
The mean is \(\boxed{17.1}\).
The variance is \(\boxed{10.5}\).
The standard deviation is \(\boxed{3.2}\).

Was this solution helpful?
failed
Unhelpful
failed
Helpful