Answer:
The 95% confidence interval for the proportion of students that obtain a letter grade of B or better from this professor is (0.2056, 0.3544). The interpretation is that we are 95% sure that the true proportion of students who obtain a letter grade of B or better from this professor is between 0.2056 and 0.3544.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence level of
, we have the following confidence interval of proportions.
![\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=%5Cpi%20%5Cpm%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
In which
z is the zscore that has a pvalue of
.
For this problem, we have that:
140 students, so ![n = 140](https://tex.z-dn.net/?f=n%20%3D%20140)
B or better are grades of A or B.
5% earn As, 23% earn Bs, so ![p = 0.05 + 0.23 = 0.28](https://tex.z-dn.net/?f=p%20%3D%200.05%20%2B%200.23%20%3D%200.28)
95% confidence level
So
, z is the value of Z that has a pvalue of
, so
.
The lower limit of this interval is:
![\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.28 - 1.96\sqrt{\frac{0.28*0.72}{140}} = 0.2056](https://tex.z-dn.net/?f=%5Cpi%20-%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.28%20-%201.96%5Csqrt%7B%5Cfrac%7B0.28%2A0.72%7D%7B140%7D%7D%20%3D%200.2056)
The upper limit of this interval is:
![\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.28 + 1.96\sqrt{\frac{0.28*0.72}{140}} = 0.3544](https://tex.z-dn.net/?f=%5Cpi%20%2B%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D%20%3D%200.28%20%2B%201.96%5Csqrt%7B%5Cfrac%7B0.28%2A0.72%7D%7B140%7D%7D%20%3D%200.3544)
The 95% confidence interval for the proportion of students that obtain a letter grade of B or better from this professor is (0.2056, 0.3544). The interpretation is that we are 95% sure that the true proportion of students who obtain a letter grade of B or better from this professor is between 0.2056 and 0.3544.