Divide 3 1/2 by 7= 1/2 a cup per person
The angles of a triangle always equal 180.
You do 180 - 80 - 45. X = 45.
Answer:
(a) The sample sizes are 6787.
(b) The sample sizes are 6666.
Step-by-step explanation:
(a)
The information provided is:
Confidence level = 98%
MOE = 0.02
n₁ = n₂ = n
![\hat p_{1} = \hat p_{2} = \hat p = 0.50\ (\text{Assume})](https://tex.z-dn.net/?f=%5Chat%20p_%7B1%7D%20%3D%20%5Chat%20p_%7B2%7D%20%3D%20%5Chat%20p%20%3D%200.50%5C%20%28%5Ctext%7BAssume%7D%29)
Compute the sample sizes as follows:
![MOE=z_{\alpha/2}\times\sqrt{\frac{2\times\hat p(1-\hat p)}{n}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%5Csqrt%7B%5Cfrac%7B2%5Ctimes%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D)
![n=\frac{2\times\hat p(1-\hat p)\times (z_{\alpha/2})^{2}}{MOE^{2}}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B2%5Ctimes%5Chat%20p%281-%5Chat%20p%29%5Ctimes%20%28z_%7B%5Calpha%2F2%7D%29%5E%7B2%7D%7D%7BMOE%5E%7B2%7D%7D)
![=\frac{2\times0.50(1-0.50)\times (2.33)^{2}}{0.02^{2}}\\\\=6786.125\\\\\approx 6787](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%5Ctimes0.50%281-0.50%29%5Ctimes%20%282.33%29%5E%7B2%7D%7D%7B0.02%5E%7B2%7D%7D%5C%5C%5C%5C%3D6786.125%5C%5C%5C%5C%5Capprox%206787)
Thus, the sample sizes are 6787.
(b)
Now it is provided that:
![\hat p_{1}=0.45\\\hat p_{2}=0.58](https://tex.z-dn.net/?f=%5Chat%20p_%7B1%7D%3D0.45%5C%5C%5Chat%20p_%7B2%7D%3D0.58)
Compute the sample size as follows:
![MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})}{n}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%5Csqrt%7B%5Cfrac%7B%5Chat%20p_%7B1%7D%281-%5Chat%20p_%7B1%7D%29%2B%5Chat%20p_%7B2%7D%281-%5Chat%20p_%7B2%7D%29%7D%7Bn%7D)
![n=\frac{(z_{\alpha/2})^{2}\times [\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})]}{MOE^{2}}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B%28z_%7B%5Calpha%2F2%7D%29%5E%7B2%7D%5Ctimes%20%5B%5Chat%20p_%7B1%7D%281-%5Chat%20p_%7B1%7D%29%2B%5Chat%20p_%7B2%7D%281-%5Chat%20p_%7B2%7D%29%5D%7D%7BMOE%5E%7B2%7D%7D)
![=\frac{2.33^{2}\times [0.45(1-0.45)+0.58(1-0.58)]}{0.02^{2}}\\\\=6665.331975\\\\\approx 6666](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2.33%5E%7B2%7D%5Ctimes%20%5B0.45%281-0.45%29%2B0.58%281-0.58%29%5D%7D%7B0.02%5E%7B2%7D%7D%5C%5C%5C%5C%3D6665.331975%5C%5C%5C%5C%5Capprox%206666)
Thus, the sample sizes are 6666.