Question
Find the mean and standard deviation for the following data set. Round to the nearest tenth.
11, 14, 15, 15, 16, 16, 17, 18, 19, 23
Use the mean and standard deviation to select the correct statement.
(1 point)
Responses
16.4 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.4.
16.4 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.4.
16.6 is a typical value for the data set. 3.7 is the typical distance between any pair of values in the data set.
16.6 is a typical value for the data set. 3.7 is the typical distance between any pair of values in the data set.
16.6 is a typical value for the data set. 3.7 is the typical distance that a value in the data set is from 16.6.
16.6 is a typical value for the data set. 3.7 is the typical distance that a value in the data set is from 16.6.
16.4 is a typical value for the data set. 3.0 is the typical distance between any pair of values in the data set.
11, 14, 15, 15, 16, 16, 17, 18, 19, 23
Use the mean and standard deviation to select the correct statement.
(1 point)
Responses
16.4 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.4.
16.4 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.4.
16.6 is a typical value for the data set. 3.7 is the typical distance between any pair of values in the data set.
16.6 is a typical value for the data set. 3.7 is the typical distance between any pair of values in the data set.
16.6 is a typical value for the data set. 3.7 is the typical distance that a value in the data set is from 16.6.
16.6 is a typical value for the data set. 3.7 is the typical distance that a value in the data set is from 16.6.
16.4 is a typical value for the data set. 3.0 is the typical distance between any pair of values in the data set.
Answers
GPT-4o mini
To find the mean and standard deviation of the data set \(11, 14, 15, 15, 16, 16, 17, 18, 19, 23\), we proceed as follows:
### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{\sum x_i}{n}
\]
Where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points.
\[
\sum x_i = 11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 14 + 15 + 15 = 29 + 16 + 16 + 17 + 18 + 19 + 23 = 30 + 17 + 18 + 19 + 23 = 47 + 18 + 19 + 23 = 65 + 19 + 23 = 84 + 23 = 107
\]
\[
n = 10
\]
Now, calculate the mean:
\[
\text{Mean} = \frac{107}{10} = 16.7
\]
### Step 2: Calculate the Standard Deviation
The standard deviation is calculated using the formula:
\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}
\]
Where \( \mu \) is the mean we calculated, \( x_i \) are the individual data points, and \( n \) is the number of data points.
1. Calculate each \((x_i - \mu)^2\):
\[
\begin{align*}
(11 - 16.7)^2 & = (-5.7)^2 = 32.49 \\
(14 - 16.7)^2 & = (-2.7)^2 = 7.29 \\
(15 - 16.7)^2 & = (-1.7)^2 = 2.89 \\
(15 - 16.7)^2 & = (-1.7)^2 = 2.89 \\
(16 - 16.7)^2 & = (-0.7)^2 = 0.49 \\
(16 - 16.7)^2 & = (-0.7)^2 = 0.49 \\
(17 - 16.7)^2 & = (0.3)^2 = 0.09 \\
(18 - 16.7)^2 & = (1.3)^2 = 1.69 \\
(19 - 16.7)^2 & = (2.3)^2 = 5.29 \\
(23 - 16.7)^2 & = (6.3)^2 = 39.69 \\
\end{align*}
\]
Now, summing these values:
\[
\sum (x_i - \mu)^2 = 32.49 + 7.29 + 2.89 + 2.89 + 0.49 + 0.49 + 0.09 + 1.69 + 5.29 + 39.69 = 92.5
\]
Now, divide by \(n\) (which is 10):
\[
\frac{92.5}{10} = 9.25
\]
Finally, take the square root to find the standard deviation:
\[
\sigma = \sqrt{9.25} \approx 3.0
\]
### Summary
- Mean = 16.7 (let's consider this 16.6 when rounding down)
- Standard deviation = 3.0
### Selection of Statement
Based on the calculated mean (approximately 16.6) and standard deviation (approximately 3.0):
The correct statement is:
**16.6 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.6.**
### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{\sum x_i}{n}
\]
Where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points.
\[
\sum x_i = 11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 14 + 15 + 15 = 29 + 16 + 16 + 17 + 18 + 19 + 23 = 30 + 17 + 18 + 19 + 23 = 47 + 18 + 19 + 23 = 65 + 19 + 23 = 84 + 23 = 107
\]
\[
n = 10
\]
Now, calculate the mean:
\[
\text{Mean} = \frac{107}{10} = 16.7
\]
### Step 2: Calculate the Standard Deviation
The standard deviation is calculated using the formula:
\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}
\]
Where \( \mu \) is the mean we calculated, \( x_i \) are the individual data points, and \( n \) is the number of data points.
1. Calculate each \((x_i - \mu)^2\):
\[
\begin{align*}
(11 - 16.7)^2 & = (-5.7)^2 = 32.49 \\
(14 - 16.7)^2 & = (-2.7)^2 = 7.29 \\
(15 - 16.7)^2 & = (-1.7)^2 = 2.89 \\
(15 - 16.7)^2 & = (-1.7)^2 = 2.89 \\
(16 - 16.7)^2 & = (-0.7)^2 = 0.49 \\
(16 - 16.7)^2 & = (-0.7)^2 = 0.49 \\
(17 - 16.7)^2 & = (0.3)^2 = 0.09 \\
(18 - 16.7)^2 & = (1.3)^2 = 1.69 \\
(19 - 16.7)^2 & = (2.3)^2 = 5.29 \\
(23 - 16.7)^2 & = (6.3)^2 = 39.69 \\
\end{align*}
\]
Now, summing these values:
\[
\sum (x_i - \mu)^2 = 32.49 + 7.29 + 2.89 + 2.89 + 0.49 + 0.49 + 0.09 + 1.69 + 5.29 + 39.69 = 92.5
\]
Now, divide by \(n\) (which is 10):
\[
\frac{92.5}{10} = 9.25
\]
Finally, take the square root to find the standard deviation:
\[
\sigma = \sqrt{9.25} \approx 3.0
\]
### Summary
- Mean = 16.7 (let's consider this 16.6 when rounding down)
- Standard deviation = 3.0
### Selection of Statement
Based on the calculated mean (approximately 16.6) and standard deviation (approximately 3.0):
The correct statement is:
**16.6 is a typical value for the data set. 3.0 is the typical distance that a value in the data set is from 16.6.**