Question
4. Select the list of data items that are included within one standard deviation of the mean from this data set.
*
Hint: List ALL (list repeated values, too) of the data values from the original data set that fall within the range of values chosen on Question 3 above.
Data set: 34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
A. 56,60,65,70,70,70,70,72,77,77,80,82,86,86
B. 34,45,50,89,95,98
C. 56,60,65,70,72,77,80,82,86
D. 34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
*
Hint: List ALL (list repeated values, too) of the data values from the original data set that fall within the range of values chosen on Question 3 above.
Data set: 34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
A. 56,60,65,70,70,70,70,72,77,77,80,82,86,86
B. 34,45,50,89,95,98
C. 56,60,65,70,72,77,80,82,86
D. 34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
Answers
GPT-4o mini
To determine which data items fall within one standard deviation of the mean from the given dataset, we first need to calculate the mean and standard deviation.
### Given data set:
34, 45, 50, 56, 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86, 89, 95, 98
### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of all data values}}{\text{Number of data values}} = \frac{34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98}{20}
\]
Calculating the sum:
\[
34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1316
\]
Now calculate the mean:
\[
\text{Mean} = \frac{1316}{20} = 65.8
\]
### Step 2: Calculate the Standard Deviation
First, we find the variance, which is the average of the squared differences from the mean.
\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N}
\]
Calculating \((x_i - \text{Mean})^2\):
\[
(34 - 65.8)^2 = 1044.64 \\
(45 - 65.8)^2 = 435.24 \\
(50 - 65.8)^2 = 252.84 \\
(56 - 65.8)^2 = 96.04 \\
(60 - 65.8)^2 = 33.64 \\
(65 - 65.8)^2 = 0.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(72 - 65.8)^2 = 37.44 \\
(77 - 65.8)^2 = 124.96 \\
(77 - 65.8)^2 = 124.96 \\
(80 - 65.8)^2 = 201.64 \\
(82 - 65.8)^2 = 262.44 \\
(86 - 65.8)^2 = 409.64 \\
(86 - 65.8)^2 = 409.64 \\
(89 - 65.8)^2 = 552.64 \\
(95 - 65.8)^2 = 847.84 \\
(98 - 65.8)^2 = 1044.64
\]
Now calculate the sum of squared differences:
\[
1044.64 + 435.24 + 252.84 + 96.04 + 33.64 + 0.64 + 17.64 + 17.64 + 17.64 + 17.64 + 37.44 + 124.96 + 124.96 + 201.64 + 262.44 + 409.64 + 409.64 + 552.64 + 847.84 + 1044.64 = 3951.6
\]
Calculating the variance:
\[
\text{Variance} = \frac{3951.6}{20} = 197.58
\]
Calculating the standard deviation:
\[
\text{Standard deviation} = \sqrt{197.58} \approx 14.04
\]
### Step 3: Define One Standard Deviation Range from the Mean
The range within one standard deviation of the mean is:
\[
\text{Mean} \pm \text{Standard deviation} = 65.8 \pm 14.04
\]
So, the range is:
\[
(65.8 - 14.04, 65.8 + 14.04) = (51.76, 79.84)
\]
### Step 4: Select values within the range (51.76, 79.84)
Now we filter the original data for items in that range:
**Values in the range (51.76, 79.84)**:
- 56
- 60
- 65
- 70
- 70
- 70
- 70
- 72
- 77
- 77
- 80
The items that fall within this range are:
**56, 60, 65, 70, 70, 70, 70, 72, 77, 77**
### Answering the Question
From the answer choices, the correct choice is:
**C. 56,60,65,70,72,77,80,82,86** (with a slight discrepancy with 80 and 82 being out of range)
Hence, **A** is the closest match:
- A. 56,60,65,70,70,70,70,72,77,77 (correct, as it falls under one standard deviation limit).
So the correct answer is **A**.
### Given data set:
34, 45, 50, 56, 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86, 89, 95, 98
### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of all data values}}{\text{Number of data values}} = \frac{34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98}{20}
\]
Calculating the sum:
\[
34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1316
\]
Now calculate the mean:
\[
\text{Mean} = \frac{1316}{20} = 65.8
\]
### Step 2: Calculate the Standard Deviation
First, we find the variance, which is the average of the squared differences from the mean.
\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N}
\]
Calculating \((x_i - \text{Mean})^2\):
\[
(34 - 65.8)^2 = 1044.64 \\
(45 - 65.8)^2 = 435.24 \\
(50 - 65.8)^2 = 252.84 \\
(56 - 65.8)^2 = 96.04 \\
(60 - 65.8)^2 = 33.64 \\
(65 - 65.8)^2 = 0.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(70 - 65.8)^2 = 17.64 \\
(72 - 65.8)^2 = 37.44 \\
(77 - 65.8)^2 = 124.96 \\
(77 - 65.8)^2 = 124.96 \\
(80 - 65.8)^2 = 201.64 \\
(82 - 65.8)^2 = 262.44 \\
(86 - 65.8)^2 = 409.64 \\
(86 - 65.8)^2 = 409.64 \\
(89 - 65.8)^2 = 552.64 \\
(95 - 65.8)^2 = 847.84 \\
(98 - 65.8)^2 = 1044.64
\]
Now calculate the sum of squared differences:
\[
1044.64 + 435.24 + 252.84 + 96.04 + 33.64 + 0.64 + 17.64 + 17.64 + 17.64 + 17.64 + 37.44 + 124.96 + 124.96 + 201.64 + 262.44 + 409.64 + 409.64 + 552.64 + 847.84 + 1044.64 = 3951.6
\]
Calculating the variance:
\[
\text{Variance} = \frac{3951.6}{20} = 197.58
\]
Calculating the standard deviation:
\[
\text{Standard deviation} = \sqrt{197.58} \approx 14.04
\]
### Step 3: Define One Standard Deviation Range from the Mean
The range within one standard deviation of the mean is:
\[
\text{Mean} \pm \text{Standard deviation} = 65.8 \pm 14.04
\]
So, the range is:
\[
(65.8 - 14.04, 65.8 + 14.04) = (51.76, 79.84)
\]
### Step 4: Select values within the range (51.76, 79.84)
Now we filter the original data for items in that range:
**Values in the range (51.76, 79.84)**:
- 56
- 60
- 65
- 70
- 70
- 70
- 70
- 72
- 77
- 77
- 80
The items that fall within this range are:
**56, 60, 65, 70, 70, 70, 70, 72, 77, 77**
### Answering the Question
From the answer choices, the correct choice is:
**C. 56,60,65,70,72,77,80,82,86** (with a slight discrepancy with 80 and 82 being out of range)
Hence, **A** is the closest match:
- A. 56,60,65,70,70,70,70,72,77,77 (correct, as it falls under one standard deviation limit).
So the correct answer is **A**.