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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

A large cable company reports that \(80 \%\) of its customers subscribe to its cable TV service, \(42 \%\) subscribe to its Internet service, and \(97 \%\) subscribe to at least one of these two services. a. Use the given probability information to set up a hypothetical 1000 table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer subscribes to both cable TV and Internet service. ii. the probability that a randomly selected customer subscribes to exactly one of these services.

The report "Twitter in Higher Education: Usage Habits and Trends of Today's College Faculty" describes a survey of nearly 2000 college faculty. The report indicates the following: \(30.7 \%\) reported that they use Twitter, and \(69.3 \%\) said that they do not use Twitter. Of those who use Twitter, \(39.9 \%\) said they sometimes use Twitter to communicate with students. Of those who use Twitter, \(27.5 \%\) said that they sometimes use Twitter as a learning tool in the classroom. Consider the chance experiment that selects one of the study participants at random. a. Two of the percentages given in the problem specify unconditional probabilities, and the other two percentages specify conditional probabilities. Which are conditional probabilities? How can you tell? b. Suppose the following events are defined: \(T=\) event that selected faculty member uses Twitter \(C=\) event that selected faculty member sometimes uses Twitter to communicate with students \(L=\) event that selected faculty member sometimes uses Twitter as a learning tool in the classroom Use the given information to determine the following probabilities: i. \(P(T)\) iii. \(P(C \mid T)\) ii. \(P\left(T^{C}\right)\) iv. \(\quad P(L \mid T)\) c. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate \(P(C),\) the probability that the selected study participant sometimes uses Twitter to communicate with students. d. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate the probability that the selected study participant sometimes uses Twitter as a learning tool in the classroom.

The article "Scrambled Statistics: What Are the Chances of Finding Multi-Yolk Eggs?" (Significance [August 2016]: 11) gives the probability of a double-yolk egg as 0.001 . a. Give a relative frequency interpretation of this probability. b. If 5000 eggs were randomly selected, about how many double-yolk eggs would you expect to find?

An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. a. Use the given probability information to set up a hypothetical 1000 table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. ii. the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer.

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.
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