The president of a city's Downtown Council hosted a meeting with 800 participants on January 27, 2005, and was curious about how many participants would have a birthday that day.
a. If the probability of having a birthday on that day for each individual is 1/365, what is the mean number of January 27 birthdays that we expect to see in a group of 800 people?
b. What is the standard deviation of the number of birthdays on January 27?
c. The council president discovered that 1 of the 800 participants was born on January 27. Find the z-score for this value.
d. Based on the z-score, is 1 a surprisingly low number of January 27 birthdays?
e. Explain why the sample size is not large enough to guarantee an approximate normal distribution.
f. According to the 68-95-99.7 Rule, the probability would be 0.95 that the number of birthdays on January 27 falls between what two values?
g. The actual binomial probability of more than 5 birthdays on January 27 in a group of 800 people can be shown to be 0.02. Explain why this is fairly consistent with the 68-95-99.7 Rule.
h. The actual probability of fewer than 0 birthdays on January 27 is, of course, zero. Explain why this is not consistent with the 68-95-99.7 Rule.