Certainly! Let's go through the problem step-by-step:
Given data set: [tex]\(10, 30, 40, 50, 60, p, 90\)[/tex]
Average of this data set: 50
### (a) Calculating the Number of Items [tex]\(n\)[/tex]
The formula for the average [tex]\(\bar{x}\)[/tex] is given by:
[tex]\[ \bar{x} = \frac{\Sigma x}{n} \][/tex]
Where [tex]\(\Sigma x\)[/tex] is the sum of all items and [tex]\(n\)[/tex] is the number of items.
Rearranging this formula to find [tex]\(n\)[/tex] we get:
[tex]\[ n = \frac{\Sigma x}{\bar{x}} \][/tex]
### (b) Determining the Total Number of Data Items
Counting the items in the given data set:
[tex]\[ 10, 30, 40, 50, 60, p, 90 \][/tex]
There are a total of 7 items including the unknown [tex]\(p\)[/tex]. So, [tex]\( n = 7 \)[/tex].
### (c) Finding the Value of [tex]\(p\)[/tex]
Given the average of the data set is 50, we use the average formula:
[tex]\[ \bar{x} = \frac{\Sigma x}{n} \][/tex]
Substitute the known values:
[tex]\[ 50 = \frac{10 + 30 + 40 + 50 + 60 + p + 90}{7} \][/tex]
Let's find the sum of the known values first:
[tex]\[ 10 + 30 + 40 + 50 + 60 + 90 = 280 \][/tex]
Now, substituting this into the equation, we have:
[tex]\[ 50 = \frac{280 + p}{7} \][/tex]
Multiply both sides by 7 to clear the fraction:
[tex]\[ 350 = 280 + p \][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[ p = 350 - 280 \][/tex]
[tex]\[ p = 70 \][/tex]
### (d) New Average After Adding 20 and 80
The new data set will be:
[tex]\[ 10, 30, 40, 50, 60, 70, 90, 20, 80 \][/tex]
The total number of items now is:
[tex]\[ n = 9 \][/tex]
We need to find the sum of the new data set:
[tex]\[ \Sigma x = 10 + 30 + 40 + 50 + 60 + 70 + 90 + 20 + 80 \][/tex]
Calculating the sum:
[tex]\[ \Sigma x = 450 \][/tex]
Now we find the new average:
[tex]\[ \text{New Average} = \frac{\Sigma x}{n} \][/tex]
[tex]\[ \text{New Average} = \frac{450}{9} \][/tex]
[tex]\[ \text{New Average} = 50 \][/tex]
So, the detailed step-by-step solutions are:
1. There are 7 items in the original data set.
2. The total number of data items is [tex]\(7\)[/tex].
3. The value of [tex]\(p\)[/tex] is [tex]\(70\)[/tex].
4. After adding 20 and 80 to the data, the new average remains [tex]\(50\)[/tex].
