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Chapter 5: Problem 16

Suppose you want to estimate the probability that a randomly selected customer at a particular grocery store will pay by credit card. Over the past 3 months, 80,500 purchases were made, and 37,100 of them were paid for by credit card. What is the estimated probability that a randomly selected customer will pay by credit card?

Short Answer

Expert verified
The estimated probability that a randomly selected customer will pay by credit card is approximately 46.09%.

Step by step solution

01

Note down the given numbers

We are given that there were 37,100 credit card purchases and a total of 80,500 purchases. So, let's denote the number of credit card purchases as CC and the total number of purchases as TP. CC = 37,100 TP = 80,500
02

Calculate the proportion of credit card purchases

To find the estimated probability that a randomly selected customer will pay by credit card, we need to divide the number of credit card purchases by the total number of purchases: Probability = \( \frac{CC}{TP} \)
03

Plug in the given numbers and calculate the probability

Now we simply substitute the given numbers into the formula: Probability = \( \frac{37,100}{80,500} \) By calculating this, we find: Probability = 0.4609 (rounded to four decimal places)
04

Express the probability as a percentage

To express the probability as a percentage, we simply multiply the decimal probability by 100: Percentage = Probability 脳 100 Percentage = 0.4609 脳 100 = 46.09% So, the estimated probability that a randomly selected customer will pay by credit card is approximately 46.09%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Credit Card Payments
Credit card payments represent a significant portion of transactions in many retail environments. In our exercise, we focus on the estimation of the likelihood that a customer will choose to pay using a credit card at a grocery store. This form of payment is popular due to its convenience and security. Credit card payments allow customers to purchase goods without the immediate need for cash.

It's important to consider factors that can affect a customer's choice of payment method, such as the store's location, the demographics of its customers, promotional offers, and more. Understanding these factors can provide businesses with insights into consumer behavior and trends, helping them plan more effectively and provide better service.
Estimate Probability
Estimating probability is an essential concept in statistics that helps us predict the likelihood of an event occurring. In the context of our exercise, we're interested in the probability that a customer at a grocery store will pay with a credit card. This helps in making decisions related to business strategy and customer management.

To estimate probability, we use a simple mathematical formula. We divide the number of successful events (in this case, payments made by credit card) by the total number of events (total purchases). The formula is:
\[ Probability = \frac{CC}{TP} \]where CC is the number of credit card payments and TP is the total number of payments.

Calculating this simple ratio gives us an insight into customer behavior patterns regarding payment methods. Understanding probabilities can be pivotal for businesses in forecasting and resource allocation.
Proportion Calculation
Proportion calculation is a straightforward yet powerful tool used to determine the relative frequency of a particular occurrence within a larger context. In our exercise, we calculate the proportion of credit card payments out of total purchases.

A proportion can be calculated using the formula:Proportion = \( \frac{Part}{Whole} \)
In this context, the part is the number of credit card transactions (37,100), and the whole is the total number of purchases (80,500).

By dividing these numbers, we find the proportion: \( \frac{37,100}{80,500} = 0.4609 \). This means that approximately 46.09% of all transactions were completed using a credit card.

Understanding proportions is crucial as it allows businesses to interpret data in a way that is intuitive and easy to communicate, making it easier to identify trends and patterns in consumer behavior.

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Most popular questions from this chapter

Consider a chance experiment that consists of selecting a customer at random from all people who purchased a car at a large car dealership during 2016 . a. In the context of this chance experiment, give an example of two events that would be mutually exclusive. b. In the context of this chance experiment, give an example of two events that would not be mutually exclusive.

a. Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.64\) and \(P(F)=0.17\) i. What is the value of \(P(E \cap F)\) ? ii. What is the value of \(P(E \cup F)\) ? b. Suppose that \(A\) and \(B\) are events with \(P(A)=0.3, P(B)=0.5\), and \(P(A \cap B)=0.15 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell? c. Suppose that \(A\) and \(B\) are events with \(P(A)=0.65\) and \(P(B)=0.57 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell?

Phoenix is a hub for a large airline. Suppose that on a particular day, 8000 passengers arrived in Phoenix on this airline. Phoenix was the final destination for 1800 of these passengers. The others were all connecting to flights to other cities. On this particular day, several inbound flights were late, and 480 passengers missed their connecting flight. Of these 480 passengers, 75 were delayed overnight and had to spend the night in Phoenix. Consider the chance experiment of choosing a passenger at random from these 8000 passengers. Calculate the following probabilities: a. the probability that the selected passenger had Phoenix as a final destination. b. the probability that the selected passenger did not have Phoenix as a final destination. c. the probability that the selected passenger was connecting and missed the connecting flight. d. the probability that the selected passenger was a connecting passenger and did not miss the connecting flight. e. the probability that the selected passenger either had Phoenix as a final destination or was delayed overnight in Phoenix. f. An independent customer satisfaction survey is planned. Fifty passengers selected at random from the 8000 passengers who arrived in Phoenix on the day described above will be contacted for the survey. The airline knows that the survey results will not be favorable if too many people who were delayed overnight are included. Write a few sentences explaining whether or not you think the airline should be worried, using relevant probabilities to support your answer.

A large cable company reports the following: \(80 \%\) of its customers subscribe to cable TV service \(42 \%\) of its customers subscribe to Internet service \(32 \%\) of its customers subscribe to telephone service \(25 \%\) of its customers subscribe to both cable TV and Internet service \(21 \%\) of its customers subscribe to both cable TV and phone service \(23 \%\) of its customers subscribe to both Internet and phone service \(15 \%\) of its customers subscribe to all three services Consider the chance experiment that consists of selecting one of the cable company customers at random. In Exercise \(5.53,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. \(P(\) cable TV only) b. \(P\) (Internet \(\mid\) cable TV) c. \(P(\) exactly two services \()\) d. \(P\) (Internet and cable TV only)

Roulette is a game of chance that involves spinning a wheel that is divided into 38 segments of equal size, as shown in the accompanying picture. A metal ball is tossed into the wheel as it is spinning, and the ball eventually lands in one of the 38 segments. Each segment has an associated color. Two segments are green. Half of the other 36 segments are red, and the others are black. When a balanced roulette wheel is spun, the ball is equally likely to land in any one of the 38 segments. a. When a balanced roulette wheel is spun, what is the probability that the ball lands in a red segment? b. In the roulette wheel shown, black and red segments alternate. Suppose instead that all red segments were grouped together and that all black segments were together. Does this increase the probability that the ball will land in a red segment? Explain. c. Suppose that you watch 1000 spins of a roulette wheel and note the color that results from each spin. What would be an indication that the wheel was not balanced?
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