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Sagot :
Sure, let's go through the problem step-by-step:
### Step 1: Calculate the Mean for Discrete Data
The formula to calculate the mean for discrete data is:
[tex]\[ \text{Mean} (\bar{x}) = \frac{\sum fx}{\text{Total number of observations (n)}} \][/tex]
Where:
- [tex]\( \sum fx \)[/tex] is the sum of the product of frequency (f) and the variable (x).
- [tex]\( n \)[/tex] is the total number of observations.
Given:
- [tex]\( \sum fx = 1620 \)[/tex]
- Number of observations ([tex]\( n \)[/tex]) = 36
Plugging in the values:
[tex]\[ \text{Mean} (\bar{x}) = \frac{1620}{36} = 45.0 \][/tex]
The mean of the original data is 45.0.
### Step 2: Calculate the New Mean After 6 Observations are Removed
If 6 observations are removed, we need to adjust the total number of observations and the sum of fx.
Number of observations removed = 6
New total number of observations = [tex]\( 36 - 6 = 30 \)[/tex]
To find the new sum of [tex]\( fx \)[/tex]:
- The 6 removed observations have the same mean as the original data which is 45.0 each.
- So, the total [tex]\( fx \)[/tex] for these 6 observations is [tex]\( 6 \times 45.0 = 270 \)[/tex].
New sum of [tex]\( fx \)[/tex] = [tex]\( 1620 - 270 = 1350 \)[/tex]
Now, the new mean is:
[tex]\[ \text{New Mean} = \frac{1350}{30} = 45.0 \][/tex]
### Step 3: Calculate the Percentage Increase in the Mean After Adding 180 to the Original Σfx
If 180 is added to the original sum of [tex]\( fx \)[/tex]:
- [tex]\( \sum fx \)[/tex] with addition = [tex]\( 1620 + 180 = 1800 \)[/tex]
The new mean after this addition:
[tex]\[ \text{New Mean with Addition} = \frac{1800}{36} = 50.0 \][/tex]
To find the percentage increase in the mean:
[tex]\[ \text{Percentage Increase} = \left( \frac{\text{New Mean with Addition} - \text{Original Mean}}{\text{Original Mean}} \right) \times 100 \][/tex]
Using the values:
[tex]\[ \text{Percentage Increase} = \left( \frac{50.0 - 45.0}{45.0} \right) \times 100 = \frac{5.0}{45.0} \times 100 \approx 11.111 \][/tex]
So, the mean increases by approximately 11.11% when 180 is added to the original sum of [tex]\( fx \)[/tex].
### Summary of Results
- The original mean is 45.0.
- The new mean after removing 6 observations is 45.0.
- The new mean after adding 180 to the original [tex]\( \sum fx \)[/tex] is 50.0.
- The percentage increase in the mean after adding 180 to the original [tex]\( \sum fx \)[/tex] is approximately 11.11%.
### Step 1: Calculate the Mean for Discrete Data
The formula to calculate the mean for discrete data is:
[tex]\[ \text{Mean} (\bar{x}) = \frac{\sum fx}{\text{Total number of observations (n)}} \][/tex]
Where:
- [tex]\( \sum fx \)[/tex] is the sum of the product of frequency (f) and the variable (x).
- [tex]\( n \)[/tex] is the total number of observations.
Given:
- [tex]\( \sum fx = 1620 \)[/tex]
- Number of observations ([tex]\( n \)[/tex]) = 36
Plugging in the values:
[tex]\[ \text{Mean} (\bar{x}) = \frac{1620}{36} = 45.0 \][/tex]
The mean of the original data is 45.0.
### Step 2: Calculate the New Mean After 6 Observations are Removed
If 6 observations are removed, we need to adjust the total number of observations and the sum of fx.
Number of observations removed = 6
New total number of observations = [tex]\( 36 - 6 = 30 \)[/tex]
To find the new sum of [tex]\( fx \)[/tex]:
- The 6 removed observations have the same mean as the original data which is 45.0 each.
- So, the total [tex]\( fx \)[/tex] for these 6 observations is [tex]\( 6 \times 45.0 = 270 \)[/tex].
New sum of [tex]\( fx \)[/tex] = [tex]\( 1620 - 270 = 1350 \)[/tex]
Now, the new mean is:
[tex]\[ \text{New Mean} = \frac{1350}{30} = 45.0 \][/tex]
### Step 3: Calculate the Percentage Increase in the Mean After Adding 180 to the Original Σfx
If 180 is added to the original sum of [tex]\( fx \)[/tex]:
- [tex]\( \sum fx \)[/tex] with addition = [tex]\( 1620 + 180 = 1800 \)[/tex]
The new mean after this addition:
[tex]\[ \text{New Mean with Addition} = \frac{1800}{36} = 50.0 \][/tex]
To find the percentage increase in the mean:
[tex]\[ \text{Percentage Increase} = \left( \frac{\text{New Mean with Addition} - \text{Original Mean}}{\text{Original Mean}} \right) \times 100 \][/tex]
Using the values:
[tex]\[ \text{Percentage Increase} = \left( \frac{50.0 - 45.0}{45.0} \right) \times 100 = \frac{5.0}{45.0} \times 100 \approx 11.111 \][/tex]
So, the mean increases by approximately 11.11% when 180 is added to the original sum of [tex]\( fx \)[/tex].
### Summary of Results
- The original mean is 45.0.
- The new mean after removing 6 observations is 45.0.
- The new mean after adding 180 to the original [tex]\( \sum fx \)[/tex] is 50.0.
- The percentage increase in the mean after adding 180 to the original [tex]\( \sum fx \)[/tex] is approximately 11.11%.
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