1. Suppose we are interested to estimate the proportion (p) of smokers among the male students in IIMA. Suppose a sample of size 100 is chosen out of the male students and the sample proportion of smokers is found to be 0.2. Give an interval based on your data so that you are 95% confident that the true value of the unknown proportion lies inside it.
How would you explain 95% confidence to a layman? Suppose a professor of IIMA thinks that true proportion is 0.3. Are you ready to accept the professor’s perception based on your data at 99% confidence level?

Solution – 1
Sample Size n = 100 (male smokers) p = 0.2
Sd (P) = √(pq / n) = .04
95% confidence interval of p
= 0.2 ± 2 x 0.04
= 0.08 to 0.32

Explanation to a layman – 95% confidence means that if the sampling experiment i.e. selection of random samples of 100 male smokers in the present problem, is repeated large no of times, 95% of the times the interval will include the true value of p (0.2) or the sample proportion of smokers in present example and 5% of the times the interval may not include the true value of p (0.2) or the sample proportion of smokers in present example.

99 % confidence interval of p
= 0.2 ± 2.58 x .04
= .04 to 0.35

Based on our data, as the true proportion of 0.3 thought by the IIMA professor lies in the 99% confidence interval, we can accept IIMA Professor’s perception.

2. A week before presidential election in USA, suppose a news agency wants to predict who is going to win (Democrat or Republican)? In order to predict, they collected a random sample of size 10000 individuals nationally and found 5100 of them intend to vote for Democrat. Based on the data could you claim with 95% confidence that Democrat would win? Could you claim the same if the confidence level be increased to 99%?

Solution – 2
Sample Size = 10,000 (National Individuals)
p