It is estimated that 62% of television viewers "channel surf" during commercial. A market research firm surveyed 1500 viewers. a) What is the probability that at least 940 of them were channel surfing?
To find the probability that at least 940 viewers were channel surfing, we can use the binomial distribution formula.
The binomial distribution formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes
C(n, k) is the number of combinations of n items taken k at a time
p is the probability of success
n is the number of trials
k is the number of successes
In this case:
p = 0.62 (probability of channel surfing)
n = 1500 (sample size)
k can range from 940 to 1500
To find the probability of at least 940 viewers channel surfing, we need to sum up the probabilities of 940, 941, 942, ..., 1500 viewers channel surfing.
P(at least 940) = P(X = 940) + P(X = 941) + P(X = 942) + ... + P(X = 1500)
We can calculate each individual probability using the binomial distribution formula.
P(at least 940) = ∑[k = 940 to 1500] ( C(1500, k) * 0.62^k * (1 - 0.62)^(1500 - k) )
However, calculating this sum manually can be quite tedious and time-consuming. We can use technology such as statistical software or online calculators to calculate the probability easily and accurately.
Please note that the formula assumes the events are independent, and the probability of channel surfing remains constant throughout the sample.