Respuesta :
Answer:
The total number of interviews will be about 9552, give or take 37.4 or so.
Step-by-step explanation:
Data given
Total of 50000 households
[tex]\bar X = 2.8[/tex] represent the average number of people in household
[tex]s= 1.87[/tex] represent the sample standard deviation of the number of people in household
n= 400 represent the sample size.
Solution to the problem
We can assume that the sample of 400 households are selected using random sampling from a grand box of 50000, and we select 400.
The expected value for the average number of people in a household is 2.38 and the standard deviation is 1.87.
We ar einterested on this case on the expected value for the sum and is defined as:
Expected value sum = Number selected * Expected value= 400*2.38=952
The reason of this is this one: From the definition of sample mean we have:
[tex]\bar X =\frac{\sum_{i=1}^n X_i}{n} =\frac{T}{n}[/tex]
Where T represent the sum or the total, if we solve for T we got:
[tex]T = n \bar x[/tex]
Assuming that [tex]X\sim N(\mu, \sigma)[/tex] and if we find the expected value for the total and the variance we got:
[tex]E(T) = n E (\bar X) = n \mu[/tex]
[tex]Var(T) = n^2 Var(T) = n^2 \sigma^2[/tex]
And the deviation would be:
[tex]Sd(X) = \sqrt{n^2 \frac{\sigma^2}{n}}= \sqrt{n} \sigma [/tex]
And now the standard error for the sum is defined as the product of the square root of n and the single deviation of a box like this:
[tex]SE = \sqrt{n} s= \sqrt{400} 1.87 =37.4[/tex]
So then the correct answer would be:
The total number of interviews will be about 9552, give or take 37.4 or so.
