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Survey: Customer Loyalty Are customers more loyal in the east or in the west? The following table is based on information from Trends in the United States, published by the Food Marketing Institute, Washington, D.C. The columns represent length of customer loyalty (in years) at a primary supermarket. The rows represent regions of the United States. $$\begin{array}{lccccccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { 1 Year } \end{array} & \begin{array}{c} 1-2 \\ \text { Years } \end{array} & \begin{array}{c} 3-4 \\ \text { Years } \end{array} & \begin{array}{c} 5-9 \\ \text { Years } \end{array} & \begin{array}{c} 10-14 \\ \text { Years } \end{array} & \begin{array}{c} 15 \text { or More } \\ \text { Years } \end{array} & \begin{array}{c} \text { Row } \\ \text { Total } \end{array} \\ \hline \text { East } & 32 & 54 & 59 & 112 & 77 & 118 & 452 \\ \text { Midwest } & 31 & 68 & 68 & 120 & 63 & 173 & 523 \\ \text { South } & 53 & 92 & 93 & 158 & 106 & 158 & 660 \\ \text { West } & 41 & 56 & 67 & 78 & 45 & 86 & 373 \\ \text { Column Total } & 157 & 270 & 287 & 468 & 291 & 535 & 2008 \\ \hline \end{array}$$ What is the probability that a customer chosen at random (a) has been loyal 10 to 14 years? (b) has been loyal 10 to 14 years, given that he or she is from the east? (c) has been loyal at least 10 years? (d) has been loyal \(a t\) least 10 years, given that he or she is from the west? (e) is from the west, given that he or she has been loyal less than I year? (f) is from the south, given that he or she has been loyal less than 1 year? (g) has been loyal I or more years, given that he or she is from the east? (h) has been loyal I or more years, given that he or she is from the west? (i) Are the events "from the east" and "loyal 15 or more years" independent? Explain.

Step by step solution

01

Probability of Loyalty 10 to 14 Years

To find the probability that a customer has been loyal for 10 to 14 years, we divide the number of customers who have been loyal for 10 to 14 years by the total number of customers. This is \( \frac{291}{2008} \approx 0.145 \).

02

Probability of Loyalty 10 to 14 Years Given East

To find the probability that a customer has been loyal for 10 to 14 years given they are from the east, divide the number of customers from the east who have been loyal for 10 to 14 years by the total number of customers from the east. This is \( \frac{77}{452} \approx 0.1704 \).

03

Probability of Loyalty at Least 10 Years

To find the probability that a customer has been loyal at least 10 years, add the totals of customers loyal for 10 to 14 years and more than 15 years, then divide by the total number of customers. This is \( \frac{291 + 535}{2008} = \frac{826}{2008} \approx 0.4113 \).

04

Probability of Loyalty at Least 10 Years Given West

To find the probability that a customer has been loyal at least 10 years given they are from the west, sum the number of Western customers loyal for 10 to 14 years and more than 15 years, then divide by the total number of Western customers. This is \( \frac{45 + 86}{373} = \frac{131}{373} \approx 0.3517 \).

05

Probability of Being from West Given Loyalty Less Than 1 Year

The probability that a customer is from the west given loyalty less than 1 year is calculated by dividing the number of Western customers loyal less than 1 year by the total number of customers loyal less than 1 year. This is \( \frac{41}{157} \approx 0.2611 \).

06

Probability of Being from South Given Loyalty Less Than 1 Year

The probability that a customer is from the south given loyalty less than 1 year is calculated by dividing the number of Southern customers loyal less than 1 year by the total number of customers loyal less than 1 year. This is \( \frac{53}{157} \approx 0.3376 \).

07

Probability of Loyalty 1 or More Years Given East

To find the probability that a customer has been loyal for 1 or more years given they are from the east, subtract the number of customers from the east loyal less than 1 year from the total number of Eastern customers, then divide by total Eastern customers. This is \( \frac{452 - 32}{452} = \frac{420}{452} \approx 0.9292 \).

08

Probability of Loyalty 1 or More Years Given West

To find the probability that a customer has been loyal for 1 or more years given they are from the west, subtract the number of Western customers loyal less than 1 year from the total number of Western customers, then divide by total Western customers. This is \( \frac{373 - 41}{373} = \frac{332}{373} \approx 0.8903 \).

09

Independence Check for East and Loyalty Over 15 Years

To check independence, compare the probability of loyalty over 15 years for Eastern customers to the overall probability. East loyalty over 15 years is \( \frac{118}{452} \approx 0.2611 \), while overall is \( \frac{535}{2008} \approx 0.2665 \). Since these probabilities are roughly equal, the events can be considered independent.

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Key Concepts

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

Conditional Probability

Conditional probability is a fundamental concept in probability and statistics. It refers to the probability of an event occurring given that another event has already occurred. For example, in the context of the survey on customer loyalty, one might find the probability that a customer has been loyal for a certain number of years given they are from a specific region, say the East or West.

Conditional probability is mathematically expressed as \( P(A|B) \), which is read as "the probability of A given B." The formula is \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \(P(A \cap B)\) is the probability of both events A and B happening, and \(P(B)\) is the probability of the event B. This concept helps focus on a subset of the population, enabling more precise predictions.

In our exercise, calculating the probability of a customer being loyal for 10 to 14 years given they are from the East involves dividing the number of Eastern customers loyally in that range by the total Eastern customers. Such calculations allow for targeted insights into customer behavior, crucial for marketers and businesses alike.

Independence of Events

Independence of events is a key topic in probability theory, playing a major role in determining whether the occurrence of one event affects another. Two events, A and B, are considered independent if the occurrence of A does not influence the probability of B occurring, and vice versa. Mathematically, this is expressed as \( P(A \cap B) = P(A) \cdot P(B) \).

In our exercise, we are examining if being from the "East" and being loyal for "15 or more years" are independent. We do this by comparing two probabilities: the overall probability of being loyal for 15 or more years and the specific probability for Eastern customers within the same loyalty span.

If these probabilities are equivalent or close, we may conclude that the two events are independent. This is critical, as it informs us that regionality does not dictate loyalty duration, allowing marketing strategies to be generalized across different regions.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of possibilities. It is the ratio of the favorable outcomes to the total number of possible outcomes. In the context of our customer loyalty survey, calculating the probability helps businesses understand customer behaviors across different regions and loyalty lengths.

For instance, finding the probability of customers being loyal for "10 to 14 years" involves dividing those who fit that loyal timeframe by the total number of surveyed customers. Similarly, when calculating the likelihood of being loyal for "at least 10 years," combine the probabilities of being loyal "10 to 14 years" and "15 or more years."

Probability calculations are essential for not just predicting outcomes, but also for strategic planning and resource allocation. They provide valuable insights into patterns and trends, enabling informed decision-making.

Contingency Tables

Contingency tables, often used in statistics, display the frequency distribution of variables. They are particularly useful for assessing the relationship between categorical variables, such as the region and loyalty span in our exercise.

A contingency table is organized with rows and columns, each representing different categories. In our example, the rows might represent regions (East, Midwest, South, West), while the columns represent loyalty spans (e.g., "Less Than 1 Year," "1-2 Years," up to "15 or More Years"). The cell in each row-column intersection represents the count of survey participants that fall into both categories.

This table structure allows us to easily perform various probability calculations, including conditional probabilities. By offering a visual representation of data, contingency tables help identify patterns and make interpreting complex data sets much more comprehensible. They are a powerful tool in analyzing and understanding multivariate data, thereby aiding in strategic decision-making.