Answer :
Sure, I'd be glad to help you with that.
### (a) Finding the Range:
The range of a set of numbers is the difference between the largest and smallest numbers in that set.
1. Identify the largest number in the set: [tex]\( \text{Largest} = 10.1 \)[/tex]
2. Identify the smallest number in the set: [tex]\( \text{Smallest} = 2.4 \)[/tex]
3. Compute the range: [tex]\(\text{Range} = \text{Largest} - \text{Smallest}\)[/tex]
So,
[tex]\[ \text{Range} = 10.1 - 2.4 = 7.7 \][/tex]
Thus, the range is 7.7.
### (b) Finding the Standard Deviation:
The standard deviation measures the amount of variation or dispersion of a set of values. Here's how to calculate it:
1. Find the mean (average) of the data set.
The given data set is: [tex]\( 8.2, 10.1, 2.6, 4.8, 2.4, 5.6, 7.0, 3.3 \)[/tex]
Sum of the data set:
[tex]\[ 8.2 + 10.1 + 2.6 + 4.8 + 2.4 + 5.6 + 7.0 + 3.3 = 44.0 \][/tex]
Number of data points (n):
[tex]\[ n = 8 \][/tex]
Mean ([tex]\(\mu\)[/tex]):
[tex]\[ \mu = \frac{\text{Sum of data}}{n} = \frac{44.0}{8} = 5.5 \][/tex]
2. Subtract the mean from each data point and square the result.
The differences from the mean [tex]\((x_i - \mu)\)[/tex] are:
[tex]\[ 8.2 - 5.5 = 2.7, \quad 10.1 - 5.5 = 4.6, \quad 2.6 - 5.5 = -2.9, \quad 4.8 - 5.5 = -0.7, \quad 2.4 - 5.5 = -3.1, \quad 5.6 - 5.5 = 0.1, \quad 7.0 - 5.5 = 1.5, \quad 3.3 - 5.5 = -2.2 \][/tex]
Squaring these differences:
[tex]\[ 2.7^2 = 7.29, \quad 4.6^2 = 21.16, \quad (-2.9)^2 = 8.41, \quad (-0.7)^2 = 0.49, \quad (-3.1)^2 = 9.61, \quad 0.1^2 = 0.01, \quad 1.5^2 = 2.25, \quad (-2.2)^2 = 4.84 \][/tex]
3. Find the mean of these squared differences.
Sum of squared differences:
[tex]\[ 7.29 + 21.16 + 8.41 + 0.49 + 9.61 + 0.01 + 2.25 + 4.84 = 54.06 \][/tex]
Mean of these squared differences (variance, [tex]\(\sigma^2\)[/tex]):
[tex]\[ \sigma^2 = \frac{54.06}{n} = \frac{54.06}{8} = 6.7575 \][/tex]
4. Take the square root of the variance to get the standard deviation ([tex]\(\sigma\)[/tex]).
[tex]\[ \sigma = \sqrt{6.7575} \approx 2.60 \][/tex]
Therefore, the standard deviation of the data set is approximately 2.60.
### Summary:
- Range = 7.7
- Standard Deviation = 2.60 (rounded to the nearest hundredth)
### (a) Finding the Range:
The range of a set of numbers is the difference between the largest and smallest numbers in that set.
1. Identify the largest number in the set: [tex]\( \text{Largest} = 10.1 \)[/tex]
2. Identify the smallest number in the set: [tex]\( \text{Smallest} = 2.4 \)[/tex]
3. Compute the range: [tex]\(\text{Range} = \text{Largest} - \text{Smallest}\)[/tex]
So,
[tex]\[ \text{Range} = 10.1 - 2.4 = 7.7 \][/tex]
Thus, the range is 7.7.
### (b) Finding the Standard Deviation:
The standard deviation measures the amount of variation or dispersion of a set of values. Here's how to calculate it:
1. Find the mean (average) of the data set.
The given data set is: [tex]\( 8.2, 10.1, 2.6, 4.8, 2.4, 5.6, 7.0, 3.3 \)[/tex]
Sum of the data set:
[tex]\[ 8.2 + 10.1 + 2.6 + 4.8 + 2.4 + 5.6 + 7.0 + 3.3 = 44.0 \][/tex]
Number of data points (n):
[tex]\[ n = 8 \][/tex]
Mean ([tex]\(\mu\)[/tex]):
[tex]\[ \mu = \frac{\text{Sum of data}}{n} = \frac{44.0}{8} = 5.5 \][/tex]
2. Subtract the mean from each data point and square the result.
The differences from the mean [tex]\((x_i - \mu)\)[/tex] are:
[tex]\[ 8.2 - 5.5 = 2.7, \quad 10.1 - 5.5 = 4.6, \quad 2.6 - 5.5 = -2.9, \quad 4.8 - 5.5 = -0.7, \quad 2.4 - 5.5 = -3.1, \quad 5.6 - 5.5 = 0.1, \quad 7.0 - 5.5 = 1.5, \quad 3.3 - 5.5 = -2.2 \][/tex]
Squaring these differences:
[tex]\[ 2.7^2 = 7.29, \quad 4.6^2 = 21.16, \quad (-2.9)^2 = 8.41, \quad (-0.7)^2 = 0.49, \quad (-3.1)^2 = 9.61, \quad 0.1^2 = 0.01, \quad 1.5^2 = 2.25, \quad (-2.2)^2 = 4.84 \][/tex]
3. Find the mean of these squared differences.
Sum of squared differences:
[tex]\[ 7.29 + 21.16 + 8.41 + 0.49 + 9.61 + 0.01 + 2.25 + 4.84 = 54.06 \][/tex]
Mean of these squared differences (variance, [tex]\(\sigma^2\)[/tex]):
[tex]\[ \sigma^2 = \frac{54.06}{n} = \frac{54.06}{8} = 6.7575 \][/tex]
4. Take the square root of the variance to get the standard deviation ([tex]\(\sigma\)[/tex]).
[tex]\[ \sigma = \sqrt{6.7575} \approx 2.60 \][/tex]
Therefore, the standard deviation of the data set is approximately 2.60.
### Summary:
- Range = 7.7
- Standard Deviation = 2.60 (rounded to the nearest hundredth)
