Sample Proportions Examples

Sample Proportions Examples

Example 1

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 1) Each new hire is a binomial experiment, with success being a minority new hire. What are the values of n and p ?

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 2) For the new hires, what is the sample proportion ?

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 3) Is the normal approximation for the distribution of appropriate? Explain.

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 4) Compare and Sample Mean and Sample Standard deviation.

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 5) Compute P(sample proportion < 0.27)

Open Ended

The general ethnic profile of Denver is about 42% minority and 58% Caucasian. Suppose the city of Denver recently hired 56 new workers. It was claimed that the hiring practice was completely unbiased regarding ethnic background. However, only 27% of the new employees are minorities, and now there is a complaint. What is the probability that at most 27% of the new hires will be minorities if the selection process is unbiased and the applicant pool reflects the ethnic profile of Denver?

Step 6) Interpret the results.

Example 2

1) A study found that 60% of all Americans believe that convicting and innocent person is worse than letting a guilty person go free. Suppose you are in a sociology class with 30 students and the questions discussed in: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? What is the probability that the proportion of the class who agree is:

Open Ended

1) A study found that 60% of all Americans believe that convicting and innocent person is worse than letting a guilty person go free. Suppose you are in a sociology class with 30 students and the questions discussed in: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? What is the probability that the proportion of the class who agree is:

a) at least one half?

Open Ended

1) A study found that 60% of all Americans believe that convicting and innocent person is worse than letting a guilty person go free. Suppose you are in a sociology class with 30 students and the questions discussed in: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? What is the probability that the proportion of the class who agree is:

b) at least two thirds?

Open Ended

1) A study found that 60% of all Americans believe that convicting and innocent person is worse than letting a guilty person go free. Suppose you are in a sociology class with 30 students and the questions discussed in: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? What is the probability that the proportion of the class who agree is:

c) no more than one third?

Open Ended

1) A study found that 60% of all Americans believe that convicting and innocent person is worse than letting a guilty person go free. Suppose you are in a sociology class with 30 students and the questions discussed in: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? What is the probability that the proportion of the class who agree is:

Is the normal approximation to the proportion valid? Explain.