Answer:
the answer should be (x+y)*(8a-1)
Answer:
The 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without is (0.500, 0.562).
Step-by-step explanation:
The complete question is:
A random sample of 1014 adults in a certain large country was asked "Do you pretty much think televisions are a necessity or a luxury you could do without?" Of the 1014 adults surveyed, 538 indicated that televisions are a luxury they could do without.Construct and interpret a 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without.
Solution:
In a sample of 1014 adults, 538 indicated that televisions are a luxury they could do without.
Compute the sample proportion of adults who indicated that televisions are a luxury they could do without as follows:
![\hat p=\frac{538}{1014}=0.531](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7B538%7D%7B1014%7D%3D0.531)
The (1 - <em>α</em>)% confidence interval for population proportion is:
![CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}](https://tex.z-dn.net/?f=CI%3D%5Chat%20p%5Cpm%20z_%7B%5Calpha%2F2%7D%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D)
The critical value of <em>z</em> for 95% confidence interval is:
![z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.05%2F2%7D%3Dz_%7B0.025%7D%3D1.96)
*use a <em>z</em>-table.
Compute the 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without as follows:
![CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}](https://tex.z-dn.net/?f=CI%3D%5Chat%20p%5Cpm%20z_%7B%5Calpha%2F2%7D%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D)
![=0.531\pm 1.96\times \sqrt{\frac{0.531(1-0.531}{1014}}}\\=0.531\pm 0.0307\\=(0.5003, 0.5617)\\\approx (0.500, 0.562)](https://tex.z-dn.net/?f=%3D0.531%5Cpm%201.96%5Ctimes%20%5Csqrt%7B%5Cfrac%7B0.531%281-0.531%7D%7B1014%7D%7D%7D%5C%5C%3D0.531%5Cpm%200.0307%5C%5C%3D%280.5003%2C%200.5617%29%5C%5C%5Capprox%20%280.500%2C%200.562%29)
Thus, the 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without is (0.500, 0.562).
Answer:
![\bold{4\sqrt3 + i4}](https://tex.z-dn.net/?f=%5Cbold%7B4%5Csqrt3%20%2B%20i4%7D)
Step-by-step explanation:
Given complex number is:
![[2(cos10^\circ + i sin10^\circ)]^3](https://tex.z-dn.net/?f=%5B2%28cos10%5E%5Ccirc%20%2B%20i%20sin10%5E%5Ccirc%29%5D%5E3)
To find:
Answer in rectangular form after using De Moivre's theorem = ?
i.e. the form
(not in forms of angles)
Solution:
De Moivre's theorem provides us a way of solving the powers of complex numbers written in polar form.
As per De Moivre's theorem:
![(cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))](https://tex.z-dn.net/?f=%28cos%5Ctheta%2Bisin%5Ctheta%29%5En%20%3D%20cos%28n%5Ctheta%29%2Bi%28sin%28n%5Ctheta%29%29)
So, the given complex number can be written as:
![[2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3](https://tex.z-dn.net/?f=%5B2%28cos10%5E%5Ccirc%20%2B%20i%20sin10%5E%5Ccirc%29%5D%5E3%5C%5C%5CRightarrow%202%5E3%20%5Ctimes%20%28cos10%5E%5Ccirc%20%2B%20i%20sin10%5E%5Ccirc%29%5E3)
Now, using De Moivre's theorem:
![\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}](https://tex.z-dn.net/?f=%5CRightarrow%202%5E3%20%5Ctimes%20%28cos10%5E%5Ccirc%20%2B%20i%20sin10%5E%5Ccirc%29%5E3%5C%5C%5CRightarrow%208%20%5Ctimes%20%5Bcos%283%20%5Ctimes10%29%5E%5Ccirc%20%2B%20i%20sin%283%20%5Ctimes10%5E%5Ccirc%29%5D%5C%5C%5CRightarrow%208%20%5Ctimes%20%28cos30%5E%5Ccirc%20%2B%20i%20sin30%5E%5Ccirc%29%5C%5C%5CRightarrow%208%20%5Ctimes%20%28%5Cdfrac%7B%5Csqrt3%7D2%20%2B%20i%20%5Cdfrac%7B1%7D%7B2%7D%29%5C%5C%5CRightarrow%20%5Cdfrac%7B%5Csqrt3%7D2%5Ctimes%208%20%2B%20i%20%5Cdfrac%7B1%7D%7B2%7D%5Ctimes%208%5C%5C%5CRightarrow%20%5Cbold%7B4%5Csqrt3%20%2B%20i4%7D)
So, the answer in rectangular form is:
![\bold{4\sqrt3 + i4}](https://tex.z-dn.net/?f=%5Cbold%7B4%5Csqrt3%20%2B%20i4%7D)
Answer:
1/2
Step-by-step explanation:
Sin 30 degree equals to the fractional value of 1/ 2. In the same way, we can derive other values of sin degrees like 0°, 30°, 45°, 60°, 90°,180°, 270° and 360°.