Question

For a particular school year, the Registrar of a University wanted to know the proportion of students who are enrolled in the Sciences. The enrollment data showed a total enrollment of 6,534 students. Of this total, there were 4,286 enrolled in various Science courses. What do the numbers say about the course preferences of the students?

A. Find the proportions p and q for each of the following:

a. X = 135, n = 378

b. X = 234, n = 512

c. X = 256, n = 624

d. X = 314, n = 850

e. X = 450, n = 1260

1.Why is p regarded as an unbiased estimator of p?

2.Describe the sample distribution of p based on large samples of size n.

Answer 1

A.

a. $\bar{x}=np, q=1-p$

$$p=\dfrac{\bar{x}}{n}=\dfrac{135}{378}=\dfrac{5}{14}$$

$$q=1-p=1-\dfrac{5}{14}=\dfrac{9}{14}$$

b. $\bar{x}=np, q=1-p$

$$p=\dfrac{\bar{x}}{n}=\dfrac{234}{512}=\dfrac{117}{256}$$

$$q=1-p=1-\dfrac{117}{256}=\dfrac{139}{256}$$

c. $\bar{x}=np, q=1-p$

$$p=\dfrac{\bar{x}}{n}=\dfrac{256}{624}=\dfrac{16}{39}$$

$$q=1-p=1-\dfrac{16}{39}=\dfrac{23}{39}$$

d. $\bar{x}=np, q=1-p$

$$p=\dfrac{\bar{x}}{n}=\dfrac{314}{850}=\dfrac{157}{425}$$

$$q=1-p=1-\dfrac{157}{425}=\dfrac{268}{425}$$

e. $\bar{x}=np, q=1-p$

$$p=\dfrac{\bar{x}}{n}=\dfrac{450}{1260}=\dfrac{5}{14}$$

$$q=1-p=1-\dfrac{5}{14}=\dfrac{9}{14}$$

1. When $X$ is a binomial rv with parameters $n$ and $p,$ the sample proportion $X/n$ is an unbiased estimator of $p.$

2. The shape of the distribution of p-hat will be approximately normal as long as the sample size n is large enough.

In practice, the approximation is adequate provided that both $np\geq 10$ and $nq\ge 10,$ since there is then enough symmetry in the underlying binomial distribution.