Grinstead and Snell’s Introduction to Probability, Chapter 9, #14

A restaurant feeds 400 customers per day. On the average 20 percent of the customers order apple pie. (a) Give a range (called a 95 percent confidence interval) for the number of pieces of apple pie ordered on a given day such that you can be 95 percent sure that the actual number will fall in this range.

n<-400
p<-0.2
stDev<-sqrt(p*(1-p)/n)
print(stDev)

## [1] 0.02

upper<-p+2*stDev
lower<-p-2*stDev

paste0("The upper value is ", upper, " and the lower value is ", lower )

## [1] "The upper value is 0.24 and the lower value is 0.16"

Therefore, there is 95% confidence that between 16-24% that apple pie was ordered.

2. How many customers must the restaurant have, on the average, to be at least 95 percent sure that the number of customers ordering pie on that day falls in the 19 to 21 percent range?

p1=.19 #lower limit
p = 0.21
q = 0.8
# .19 = .2 - 2SD
stDev= (0.21 -0.19)/2
stDev

## [1] 0.01

n <- (p*q)/ stDev^2
(n)

## [1] 1680

The restaurant will have an average of 1680 customers.