Your friends are getting married. They have to rush to send out wedding invitations because mail delivery is delayed due to the pandemic. The time it takes for mail to be delivered follows a Normal distribution with an average of 20 days and standard deviation of 4 days. The correspondences' delivery time is independent of each other. Suppose the wedding date is set to be on May 20th.

1. What is the latest day they should send an invitation so that a guest receives it on time (i.e. seven days before the wedding) with 90% probability?
2. Suppose they accidentally send an invitation late, on May 6th. What is the probability that the guest will receive the invitation before the wedding (i.e. less than or equal to 13 days)?
3. 20 guests are "almost" guaranteed to be present at the wedding (family/best people of the bride and groom). What is the latest day they should send the invitation, so that all 20 guests receive them at least seven days before the wedding with a 95% probability.
4. What is the probability that the average time of invitations sent to the 20 people involved in the wedding will be lower than 19 days? List any assumptions.
5. They want to send the rest of the invitations today. However, the postal service office closes in 4 hours and 30 minutes. Assuming they spend an average of 1 minute to prepare each invitation, with a standard deviation of 10 seconds, what is the probability they finish the remaining invitations on time? (Hint: think in terms of the mean.)
6. Suppose they will send all the 700 invitations on April 20th. What is the probability that more than 80% of their guests receive the invitation at least seven days before the wedding?
7. If they want all the 700 guests to receive their invitations at least ten days before the wedding, with 90% probability, what is the latest day they can send the invitations?
8. Suppose that guests who receive the invitation at least seven days before the wedding have a 90% probability of attending the wedding. In comparison, guests who receive the invitation within seven days of the wedding have a 75% probability of attending the wedding. Let N be the number of people that will attend to the wedding. Can we say that N follows a binomial distribution? Explain.