Answer:
(1)
Step-by-step explanation:
Data given and notation
n=100 represent the random sample taken
estimated proportion with the survey
is the value that we want to test
represent the significance level
z would represent the statistic (variable of interest)
represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the true proportion is lower than 0.41.:
Null hypothesis:
Alternative hypothesis:
When we conduct a proportion test we need to use the z statistic, and the is given by:
(1)
The One-Sample Proportion Test is used to assess whether a population proportion
is significantly different from a hypothesized value
.
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
There's no if about it,
![f(x)=x^3+3x^2-x-3 ](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E3%2B3x%5E2-x-3%0A)
has a zero
![f(1)=0](https://tex.z-dn.net/?f=f%281%29%3D0)
so
![x-1](https://tex.z-dn.net/?f=x-1)
is a factor. That's the special case of the Remainder Theorem; since
![f(1)=0](https://tex.z-dn.net/?f=f%281%29%3D0)
we'll get a remainder of zero when we divide
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
by
![x-1.](https://tex.z-dn.net/?f=x-1.)
At this point we can just divide or we can try more little numbers in the function. It doesn't take too long to discover
![f(-1)=0](https://tex.z-dn.net/?f=f%28-1%29%3D0)
too, so
![x+1](https://tex.z-dn.net/?f=x%2B1)
is a factor too by the remainder theorem. I can find the third zero as well; but let's say that's out of range for most folks.
So far we have
![x^3+3x^2-x-3 = (x-1)(x+1)(x-r)](https://tex.z-dn.net/?f=x%5E3%2B3x%5E2-x-3%20%3D%20%28x-1%29%28x%2B1%29%28x-r%29)
where
![r](https://tex.z-dn.net/?f=r)
is the zero we haven't guessed yet. Again we could divide
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
by
![(x-1)(x+1)=x^2-1](https://tex.z-dn.net/?f=%28x-1%29%28x%2B1%29%3Dx%5E2-1)
but just looking at the constant term we must have
![-3 = -1 (1)(-r) = r](https://tex.z-dn.net/?f=-3%20%3D%20-1%20%281%29%28-r%29%20%3D%20r)
so
![x^3+3x^2-x-3 = (x-1)(x+1)(x+3)](https://tex.z-dn.net/?f=x%5E3%2B3x%5E2-x-3%20%3D%20%28x-1%29%28x%2B1%29%28x%2B3%29)
We check
![f(-3)=(-3)^3+3(-3)^2 -(-3)-3 = 0 \quad\checkmark](https://tex.z-dn.net/?f=f%28-3%29%3D%28-3%29%5E3%2B3%28-3%29%5E2%20-%28-3%29-3%20%3D%200%20%5Cquad%5Ccheckmark)
We usually talk about the zeros of a function and the roots of an equation; here we have a function
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
whose zeros are
Answer:
16 and 12
Step-by-step explanation:
16+12 = 28
16-12=4