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a) In a continuous series, if [tex]\bar{X}=20, \Sigma f m=800+15p[/tex] and [tex]N=10+p[/tex], find the value of [tex]p[/tex] and the exact value of [tex]N[/tex].

Sagot :

Sure, let's solve this step-by-step together.

We are given the following values in a continuous series:
- The mean [tex]\(\bar{X} = 20\)[/tex]
- [tex]\(\Sigma fm = 800 + 15p\)[/tex]
- [tex]\(N = 10 + p\)[/tex]

We need to find the value of [tex]\(p\)[/tex] and the exact value of [tex]\(N\)[/tex].

The formula for the mean [tex]\(\bar{X}\)[/tex] in a continuous series is given by:
[tex]\[ \bar{X} = \frac{\Sigma fm}{N} \][/tex]

Substituting the known values into the formula, we get:
[tex]\[ 20 = \frac{800 + 15p}{10 + p} \][/tex]

To solve for [tex]\(p\)[/tex], we will first eliminate the fraction by multiplying both sides of the equation by [tex]\((10 + p)\)[/tex]:
[tex]\[ 20 (10 + p) = 800 + 15p \][/tex]

Next, distribute the 20 on the left side:
[tex]\[ 200 + 20p = 800 + 15p \][/tex]

Now, we will isolate [tex]\(p\)[/tex] by moving all [tex]\(p\)[/tex] terms to one side and the constants to the other side:
[tex]\[ 20p - 15p = 800 - 200 \][/tex]
[tex]\[ 5p = 600 \][/tex]

Now, divide both sides by 5 to solve for [tex]\(p\)[/tex]:
[tex]\[ p = \frac{600}{5} \][/tex]
[tex]\[ p = 120 \][/tex]

So, the value of [tex]\(p\)[/tex] is 120.

To find the exact value of [tex]\(N\)[/tex], substitute [tex]\(p\)[/tex] back into the equation for [tex]\(N\)[/tex]:
[tex]\[ N = 10 + p \][/tex]
[tex]\[ N = 10 + 120 \][/tex]
[tex]\[ N = 130 \][/tex]

Therefore, the value of [tex]\(p\)[/tex] is [tex]\(120\)[/tex], and the exact value of [tex]\(N\)[/tex] is [tex]\(130\)[/tex].