SOLUTION: Starbucks has found that 25% of customers purchase more than one cup of coffee per day from their shops.Suppose you randomly choose 200 of today's Starbucks customers.Find the prob

Question 1153010: Starbucks has found that 25% of customers purchase more than one cup of coffee per day from their shops.Suppose you randomly choose 200 of today's Starbucks customers.Find the probability that no less than 34% of them purchased more than one cup of coffee from Starbucks.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Define a success to be the event a customer purchases more than one cup of coffee per day.
p = probability of success
p = 0.25

n = 200 is the sample size

The phrasing "no less than 34% of them [the 200 sampled]" means
34% of 200 = 0.34*200 = 68
So we can rephrase "no less than 34% of them" into "no less than 68" which is equivalent to saying "at least 68"
In other words, we want to find where X = number of customers who purchase more than one cup of coffee.

The value of X ranges from 0 to 200, and it is a whole number. If X = 0, then no people bought more than one cup of coffee. If X = 1, then exactly one person bought more than one cup. Etc etc.

Normally we would use a binomial distribution (since we have independent trials and each trial is a yes/no question), but computing will be tedious through the binomial formula.

Instead, it is better to a get normal approximation.
Note that,
n*p = 200*0.25 = 50
n*(1-p) = 200*(1-0.25) = 200*0.75 = 150
are both greater than 5. Since and , we can use a normal approximation to the binomial.

Compute the mean (mu) and standard deviation (sigma)

mu = population mean
mu = n*p
mu = 200*0.25
mu = 50

sigma = sqrt(n*p*(1-p))
sigma = sqrt(200*0.25*(1-0.25))
sigma = 6.12372435695794

Instead of finding , we need to find . This is due to the Continuity Correction Factor as shown on this page
https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/
(see part 2 of that page)

Compute the z score to the raw score of x = 67.5
z = (x - mu)/sigma
z = (67.5 - 50)/6.12372435695794
z = 2.85773803324704
z = 2.86

This means

Use a table such as this one
http://www.z-table.com/
(a similar one can be found in the back of your stat textbook)
to find that
P(Z < 2.86) = 0.9979
So,
P(Z > 2.86) = 1-P(Z < 2.86)
P(Z > 2.86) = 1-0.9979
P(Z > 2.86) = 0.0021

Answer: 0.0021

RELATED QUESTIONS
Margaret manages a fast food restaurant and she is concerned about customers who ask for (answered by stanbon)

Could you please help me solve this question. I did some of it. Thank you It is not... (answered by ankor@dixie-net.com)

From a large survey of customers using a chain of coffee shops, 60% of the customers are... (answered by stanbon)

Updated questions with added information: A specific study found that the average... (answered by ikleyn)

Sung’s Coffee Shop sells a cup of tea for $1.05, a cup of coffee for $1.35, and a cup... (answered by ewatrrr,josgarithmetic)

Each week I have $11 to spend on coffee. I require 10 cups of coffee a week to survive.... (answered by josmiceli)

I can not figure out this problem, please help: In most businesses, increasing prices of... (answered by checkley71)

In most businesses, increasing prices of their product can have a negative effect on... (answered by ankor@dixie-net.com)

I need help with word problems, thank you! The caffeine content of a cup of... (answered by stanbon)