Answer:
a
The 90% confidence interval that estimate the true proportion of students who receive financial aid is
![0.533 < p < 0.64](https://tex.z-dn.net/?f=0.533%20%20%3C%20%20p%20%3C%20%200.64%20)
b
![n = 1789](https://tex.z-dn.net/?f=n%20%3D%201789)
Step-by-step explanation:
Considering question a
From the question we are told that
The sample size is n = 200
The number of student that receives financial aid is ![k = 118](https://tex.z-dn.net/?f=k%20%3D%20118)
Generally the sample proportion is
![\^ p = \frac{k}{n}](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%20%5Cfrac%7Bk%7D%7Bn%7D)
=> ![\^ p = \frac{118}{200}](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%20%5Cfrac%7B118%7D%7B200%7D)
=> ![\^ p = 0.59](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%200.59)
From the question we are told the confidence level is 90% , hence the level of significance is
![\alpha = (100 - 90 ) \%](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%28100%20-%2090%20%29%20%5C%25)
=> ![\alpha = 0.10](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.10)
Generally from the normal distribution table the critical value of
is
![Z_{\frac{\alpha }{2} } = 1.645](https://tex.z-dn.net/?f=Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%3D%20%201.645)
Generally the margin of error is mathematically represented as
![E = Z_{\frac{\alpha }{2} } * \sqrt{\frac{\^ p (1- \^ p)}{n} }](https://tex.z-dn.net/?f=E%20%3D%20%20Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%2A%20%5Csqrt%7B%5Cfrac%7B%5C%5E%20p%20%281-%20%5C%5E%20p%29%7D%7Bn%7D%20%7D%20)
=>![E = 1.645 * \sqrt{\frac{0.59 (1- 0.59)}{200} }](https://tex.z-dn.net/?f=E%20%3D%20%201.645%20%2A%20%5Csqrt%7B%5Cfrac%7B0.59%20%281-%200.59%29%7D%7B200%7D%20%7D%20)
=> ![E = 0.057](https://tex.z-dn.net/?f=E%20%3D%200.057%20)
Generally 90% confidence interval is mathematically represented as
![\^ p -E < p < \^ p +E](https://tex.z-dn.net/?f=%5C%5E%20p%20-E%20%3C%20%20p%20%3C%20%20%5C%5E%20p%20%2BE)
=>
Considering question b
From the question we are told that
The margin of error is E = 0.03
From the question we are told the confidence level is 99% , hence the level of significance is
![\alpha = (100 - 99 ) \%](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%28100%20-%2099%20%29%20%5C%25)
=> ![\alpha = 0.01](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.01)
Generally from the normal distribution table the critical value of is
![Z_{\frac{\alpha }{2} } = 2.58](https://tex.z-dn.net/?f=Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%3D%202.58)
Generally the sample size is mathematically represented as
![[\frac{Z_{\frac{\alpha }{2} }}{E} ]^2 * \^ p (1 - \^ p )](https://tex.z-dn.net/?f=%5B%5Cfrac%7BZ_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%7D%7BE%7D%20%5D%5E2%20%2A%20%5C%5E%20p%20%281%20-%20%5C%5E%20p%20%29)
=> ![n = [\frac{2.58}{0.03} ]^2 * 0.59 (1 - 0.59 )](https://tex.z-dn.net/?f=n%20%3D%20%5B%5Cfrac%7B2.58%7D%7B0.03%7D%20%5D%5E2%20%2A%200.59%20%281%20-%200.59%20%29%20)
=> ![n = 1789](https://tex.z-dn.net/?f=n%20%3D%201789)