You have the following expression:
![(1+cosx)(1-\cos x)=\sin ^2x](https://tex.z-dn.net/?f=%281%2Bcosx%29%281-%5Ccos%20x%29%3D%5Csin%20%5E2x)
To verify the identity, you have that the left side of the equation becomes:
(1 + cosx)(1 - cosx) = (1 - cos^2 x)
because of the product of an expression by its conjugate result in a difference of squares. Or simplfy by expanding the factors and simplying.
Next, you have:
1 - cos^2 x = sin^2 x
because the Pythagorean identity.
Then, the identity is verified.
Answer:
Hence, the minimum required sample size (n) should be = 1692
Step-by-step explanation:
Solution:-
- The estimation error, E = 2%
- The confidence level, CI = 90 %
- Since the proportion of the positive lab tests for prostate cancer are actually proven correct through subsequent biopsy are unknown we will assume the corresponding proportion to be p = 0.58
- The required sample size is a function of confidence value and error of estimation (E):
Where,
- The critical value of the confidence level = 90% would be:
significance level ( α ) = 1 - CI = 1 - 0.90 = 0.1
Z-critical = Z_α/2 = Z_0.05 = 1.645
- The required sample size (n) can be calculated:
![n = 0.5*( 1 - 0.5 ) * (\frac{1.645}{0.02})^2\\\\n = 1691.265](https://tex.z-dn.net/?f=n%20%3D%200.5%2A%28%201%20-%200.5%20%29%20%2A%20%28%5Cfrac%7B1.645%7D%7B0.02%7D%29%5E2%5C%5C%5C%5Cn%20%3D%201691.265)
- Hence, the minimum required sample size (n) should be = 1692
Ok, find change then divide by years elapsed
change=310-280 million=30 million
number of years elapsed=10
30mil/10=3mil
3million per year was the population growth rate