Answer:
(a) The 90% confidence interval estimate of the percentage of orders that are not accurate in Restaurant <em>A</em> is (0.148, 0.222).
(b) Restaurant <em>B </em>has more proportion of not accurate orders.
Step-by-step explanation:
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)
(a)
In Restaurant <em>A</em> the number of not accurate orders was <em>x</em> = 55 of <em>n</em> = 297 orders.
The sample proportion of not accurate orders in Restaurant <em>A</em> is:
![\hat p=\frac{x}{n}=\frac{55}{297}=0.1852](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7Bx%7D%7Bn%7D%3D%5Cfrac%7B55%7D%7B297%7D%3D0.1852)
The critical value of <em>z</em> for 90% confidence level is:
![z_{\alpha/2}=z_{0.10/2}=z_{0.05}=1.645](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.10%2F2%7D%3Dz_%7B0.05%7D%3D1.645)
Compute the 90% confidence interval estimate of the percentage of orders that are not accurate in Restaurant <em>A</em> as follows:
![CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\=0.1852\pm 1.645\sqrt{\frac{0.1852(1-0.1852)}{297}}\\=0.1852\pm 0.0371\\=(0.1481, 0.2223)\\\approx (0.148, 0.222)](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%5C%5C%3D0.1852%5Cpm%201.645%5Csqrt%7B%5Cfrac%7B0.1852%281-0.1852%29%7D%7B297%7D%7D%5C%5C%3D0.1852%5Cpm%200.0371%5C%5C%3D%280.1481%2C%200.2223%29%5C%5C%5Capprox%20%280.148%2C%200.222%29)
Thus, the 90% confidence interval estimate of the percentage of orders that are not accurate in Restaurant <em>A</em> is (0.148, 0.222).
(b)
The 90% confidence interval estimate of the percentage of orders that are not accurate in Restaurant <em>B</em> is (0.171, 0.245).
The confidence interval for Restaurant <em>B</em> indicates that between 17.1% to 24.5% orders are inaccurate.
The values of this interval is more than that for Restaurant <em>A</em>.
So, it can be concluded that Restaurant <em>B </em>has more proportion of not accurate orders.