stats 1000 sampling distribution of sample proportion

Question

Suppose that 10% of the population of college students is left-handed. A classroom has

been built with 15 “lefty seats” (e.g. seats that have the built-in desk on the left rather than the

right).

(a) What is the sampling distribution of the sample proportion of left-handed students, for a

class of 100 students (assuming a class is a random sample of students from the university)?

Answer 1

Hi Aanchal !

The exact distribution of the number of left-handed students is Binomial with n = 100 and p = 0.1. Therefore the sampling distribution of the sample proportion is (1/100) times such a random variable.

However, for large sample sizes like this typically the Normal approximation is used:



p-hat ~ N(p, p(1-p)/n)

where p-hat is the symbol for the sample proportion, p is the symbol for the population proportion (also the mean of the sampling distribution), n is the sample size and p(1-p)/n is the sampling variance.

In your case, this would mean the sampling distribution for p-hat is approximately:

p-hat ~ N(0.1, 0.0009)

Just to give you a heads' up, most textbooks and classes have validity conditions which must be met before this approximation can be used, like:

np >= 10,

n(1-p)>=10

Be sure to check your specific textbook's validity conditions are met before using the Normal approximation!