Q1) X is a random variable which follows a Binomial distribution. The mean of the distribution is 20 and the standard deviation is 4. The number of trials is 100.
a) Calculate the probability that X is equal to 15.
b) What is the probability that X is less than or equal to 62?
c) A researcher chooses a value of X called ‘A’ such that the 20% of the distribution lies below A. What is the value of A?
d) The researcher repeats the experiment for a different random variable, Y. Y is also distributed according to a Binomial distribution with mean 20 and standard deviation 4. There are 5 trials. How would your answer to part c) change? Explain as fully as possible.
Q2) A whale-watching boat goes out daily. The skipper notes that, on average, one whale is sighted during a 5 hour trip.
a) What is the probability of seeing 3 or more whales during a 5 hour long trip?
b) What is the probability that during a shorter trip of 2 hours that exactly 2 whales will be seen?
c) Some tourists see a shark during their 5 hour boat trip. They are told that for 80% of the longer boat trips, no sharks are sighted at all. What is the probability of seeing at least one shark and at least one whale during a 5 hour boat trip?
d) What assumption did you need to make to answer part c)?
e) The skipper wants to know the average number of sightings of fish eagles in one week. The probability of having no sightings of fish eagles within one day is approximately 0.75. What is the average number of fish eagles sighted in one week (to nearest whole number)?