If 55% of a television viewing population watched a program called ¡§Name that
Poem¡¨ one evening, what is the probability that, in a random sample of 100
viewers, less than 50% of the sample watched the program?
To find the probability that less than 50% of a random sample of 100 viewers watched the program, we need to use the concept of binomial distribution.
The binomial distribution calculates the probability of obtaining a specific number of successes (watching the program) in a fixed number of independent Bernoulli trials (viewers). In this case, the probability of success is 55%, and we want to find the probability of having less than 50% success in 100 trials.
To calculate this probability, we can use a binomial probability formula or a statistical software that provides binomial distribution calculators. Here, I will explain how to use the formula:
Step 1: Calculate the probability of success (p) and the probability of failure (q).
Given that 55% of the television viewing population watched the program, the probability of success (p) is 0.55, and the probability of failure (q) is 1 - p = 1 - 0.55 = 0.45.
Step 2: Determine the desired number of successes. In this case, we want to find the probability of having less than 50% success in a sample of 100 viewers.
Step 3: Sum up the probabilities for all the desired number of successes.
To find the probability of having less than 50% of the sample watching the program, we need to sum the probabilities of having 0, 1, 2, ..., 49 successes in 100 trials.
Using the binomial distribution formula, the probability of k successes in n trials is given by:
P(X = k) = (n choose k) * p^k * q^(n-k)
where (n choose k) represents the combination of n trials taken k at a time, which is calculated as n! / (k! * (n-k)!)
Step 4: Calculate the cumulative probability.
After calculating the individual probabilities for each desired number of successes, we need to sum them up. This will give us the cumulative probability of having less than 50% of the sample watching the program.
Using these steps, you can calculate the probability that less than 50% of the sample watched the program.