Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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%.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.