Answer:
![a.\ \hat p=0.2877\\\\b.\ ME=0.0263\\\\c.\ 0.2614](https://tex.z-dn.net/?f=a.%5C%20%5Chat%20p%3D0.2877%5C%5C%5C%5Cb.%5C%20ME%3D0.0263%5C%5C%5C%5Cc.%5C%200.2614%3C%5Chat%20p%3C0.3140)
d. -Sample selection was random
-Individual observations are independent of each other
np≥10
Step-by-step explanation:
a. The point estimate of a sample proportion is obtained using the formula;
![\hat p=\frac{x}{n}\\\\x-sample \ size\\n-population\ size\\\hat p-point \ estimate\\\\\\\\\therefore \hat p=\frac{328}{1140}\\\\\\=0.2877](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7Bx%7D%7Bn%7D%5C%5C%5C%5Cx-sample%20%5C%20size%5C%5Cn-population%5C%20size%5C%5C%5Chat%20p-point%20%5C%20estimate%5C%5C%5C%5C%5C%5C%5C%5C%5Ctherefore%20%5Chat%20p%3D%5Cfrac%7B328%7D%7B1140%7D%5C%5C%5C%5C%5C%5C%3D0.2877)
Hence, the point estimate of the proportion of the population is 0.2877
b. The desired margin of error is the calculated using the point estimate value as follows:
![ME=z\sqrt{\frac{\hat p(1-\hat p)}{n}}\\\\z_{0.025}=1.96\\\\\hat p=0.2877\\\\\therefore ME=1.96\times \sqrt{\frac{0.2877(1-0.2877)}{1140}} \\\\=0.0263](https://tex.z-dn.net/?f=ME%3Dz%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D%5C%5C%5C%5Cz_%7B0.025%7D%3D1.96%5C%5C%5C%5C%5Chat%20p%3D0.2877%5C%5C%5C%5C%5Ctherefore%20ME%3D1.96%5Ctimes%20%5Csqrt%7B%5Cfrac%7B0.2877%281-0.2877%29%7D%7B1140%7D%7D%20%5C%5C%5C%5C%3D0.0263)
Hence, the desired margin of error for the sample proportion is 0.0263
c. Given a confidence level of 95%, the confidence interval can be calculated as:
![CI_{95\%}=\hat p\pm ME\\\\=0.2877\pm 0.0263\\\\=[0.2614,0.3140]](https://tex.z-dn.net/?f=CI_%7B95%5C%25%7D%3D%5Chat%20p%5Cpm%20ME%5C%5C%5C%5C%3D0.2877%5Cpm%200.0263%5C%5C%5C%5C%3D%5B0.2614%2C0.3140%5D)
Hence, the confidence interval at 95% confidence level is 0.2614<p<0.3140
#We are 95% confident that the interval estimate contains the desired proportion.
d. The assumptions are:
-The sample size is is more than 10 or equal to 10:
![n\hat p\geq 10\\n\hat p=0.2877\times 328=94.37\geq 10\\\\\therefore np\geq 10](https://tex.z-dn.net/?f=n%5Chat%20p%5Cgeq%2010%5C%5Cn%5Chat%20p%3D0.2877%5Ctimes%20328%3D94.37%5Cgeq%2010%5C%5C%5C%5C%5Ctherefore%20np%5Cgeq%2010)
-The selection was from a randomized experiment.
-The individual observations were independent of each other.