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 between any pair of values in the data set.
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.
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.
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 between any pair of values in the data set.
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.
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.
Answers
GPT-4o mini
To find the mean and standard deviation of the given data set (11, 14, 15, 15, 16, 16, 17, 18, 19, 23), let's follow these steps:
### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23}{10}
\]
Calculating the sum:
\[
11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 11 + 14 = 25
\]
\[
25 + 15 = 40
\]
\[
40 + 15 = 55
\]
\[
55 + 16 = 71
\]
\[
71 + 16 = 87
\]
\[
87 + 17 = 104
\]
\[
104 + 18 = 122
\]
\[
122 + 19 = 141
\]
\[
141 + 23 = 164
\]
Now, divide by the number of data points (10):
\[
\text{Mean} = \frac{164}{10} = 16.4
\]
### Step 2: Calculate the Standard Deviation
The formula for the standard deviation is:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]
Where \( \bar{x} \) is the mean, \( x_i \) is each value, and \( n \) is the number of values.
1. Calculate each \( (x_i - \bar{x})^2 \):
- For \( 11 \): \( (11 - 16.4)^2 = (-5.4)^2 = 29.16 \)
- For \( 14 \): \( (14 - 16.4)^2 = (-2.4)^2 = 5.76 \)
- For \( 15 \): \( (15 - 16.4)^2 = (-1.4)^2 = 1.96 \)
- For \( 15 \): \( (15 - 16.4)^2 = (-1.4)^2 = 1.96 \)
- For \( 16 \): \( (16 - 16.4)^2 = (-0.4)^2 = 0.16 \)
- For \( 16 \): \( (16 - 16.4)^2 = (-0.4)^2 = 0.16 \)
- For \( 17 \): \( (17 - 16.4)^2 = (0.6)^2 = 0.36 \)
- For \( 18 \): \( (18 - 16.4)^2 = (1.6)^2 = 2.56 \)
- For \( 19 \): \( (19 - 16.4)^2 = (2.6)^2 = 6.76 \)
- For \( 23 \): \( (23 - 16.4)^2 = (6.6)^2 = 43.56 \)
2. Sum these squared differences:
\[
29.16 + 5.76 + 1.96 + 1.96 + 0.16 + 0.16 + 0.36 + 2.56 + 6.76 + 43.56 = 92.6
\]
3. Calculate the variance:
\[
\text{Variance} = \frac{92.6}{10 - 1} = \frac{92.6}{9} \approx 10.29
\]
4. Calculate the standard deviation:
\[
s = \sqrt{10.29} \approx 3.2
\]
### Summary of Results:
- **Mean**: 16.4 (rounded to the nearest tenth)
- **Standard Deviation**: 3.2 (rounded to the nearest tenth)
### Selecting the Correct Statement
The closest correct statement based on our calculated values is:
**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.**
It's worth noting that although we calculated the standard deviation to be 3.2, the closest response to that value and its context in the statements is 3.0, assuming a rounding in the option provided.
### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23}{10}
\]
Calculating the sum:
\[
11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 11 + 14 = 25
\]
\[
25 + 15 = 40
\]
\[
40 + 15 = 55
\]
\[
55 + 16 = 71
\]
\[
71 + 16 = 87
\]
\[
87 + 17 = 104
\]
\[
104 + 18 = 122
\]
\[
122 + 19 = 141
\]
\[
141 + 23 = 164
\]
Now, divide by the number of data points (10):
\[
\text{Mean} = \frac{164}{10} = 16.4
\]
### Step 2: Calculate the Standard Deviation
The formula for the standard deviation is:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]
Where \( \bar{x} \) is the mean, \( x_i \) is each value, and \( n \) is the number of values.
1. Calculate each \( (x_i - \bar{x})^2 \):
- For \( 11 \): \( (11 - 16.4)^2 = (-5.4)^2 = 29.16 \)
- For \( 14 \): \( (14 - 16.4)^2 = (-2.4)^2 = 5.76 \)
- For \( 15 \): \( (15 - 16.4)^2 = (-1.4)^2 = 1.96 \)
- For \( 15 \): \( (15 - 16.4)^2 = (-1.4)^2 = 1.96 \)
- For \( 16 \): \( (16 - 16.4)^2 = (-0.4)^2 = 0.16 \)
- For \( 16 \): \( (16 - 16.4)^2 = (-0.4)^2 = 0.16 \)
- For \( 17 \): \( (17 - 16.4)^2 = (0.6)^2 = 0.36 \)
- For \( 18 \): \( (18 - 16.4)^2 = (1.6)^2 = 2.56 \)
- For \( 19 \): \( (19 - 16.4)^2 = (2.6)^2 = 6.76 \)
- For \( 23 \): \( (23 - 16.4)^2 = (6.6)^2 = 43.56 \)
2. Sum these squared differences:
\[
29.16 + 5.76 + 1.96 + 1.96 + 0.16 + 0.16 + 0.36 + 2.56 + 6.76 + 43.56 = 92.6
\]
3. Calculate the variance:
\[
\text{Variance} = \frac{92.6}{10 - 1} = \frac{92.6}{9} \approx 10.29
\]
4. Calculate the standard deviation:
\[
s = \sqrt{10.29} \approx 3.2
\]
### Summary of Results:
- **Mean**: 16.4 (rounded to the nearest tenth)
- **Standard Deviation**: 3.2 (rounded to the nearest tenth)
### Selecting the Correct Statement
The closest correct statement based on our calculated values is:
**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.**
It's worth noting that although we calculated the standard deviation to be 3.2, the closest response to that value and its context in the statements is 3.0, assuming a rounding in the option provided.