Defining the Random Variable

Oct 13, 2025 by recoveryshake.com 10 views

Introduction

To determine whether its service is satisfactory to its customers, a hotel surveyed 100 guests. The results are summarized in the table below. A guest is randomly selected from these 100 people. The hotel wants to know the probability that a randomly selected guest is satisfied with the service. This problem can be solved using probability theory and the concept of random sampling.

Understanding the Problem

The problem involves a hotel that wants to determine the probability of a randomly selected guest being satisfied with the service. The hotel has surveyed 100 guests and obtained the results, which are summarized in the table below.

Category	Satisfied	Not Satisfied
Food	80	20
Service	70	30
Cleanliness	85	15
Overall	75	25

Defining the Random Variable

Let X be the random variable that represents the satisfaction level of a randomly selected guest. X can take on two values: 1 if the guest is satisfied, and 0 if the guest is not satisfied.

Calculating the Probability

To calculate the probability that a randomly selected guest is satisfied, we need to use the concept of random sampling. The probability of a guest being satisfied is equal to the number of guests who are satisfied divided by the total number of guests surveyed.

Using the Table to Calculate the Probability

From the table, we can see that there are 75 guests who are satisfied with the service. The total number of guests surveyed is 100. Therefore, the probability that a randomly selected guest is satisfied is:

P(X = 1) = Number of guests who are satisfied / Total number of guests surveyed = 75 / 100 = 0.75

Interpreting the Results

The probability that a randomly selected guest is satisfied with the service is 0.75. This means that there is a 75% chance that a randomly selected guest will be satisfied with the service.

Conclusion

In conclusion, the hotel can use the probability of 0.75 to determine whether its service is satisfactory to its customers. If the probability is greater than 0.5, then the service is satisfactory. If the probability is less than 0.5, then the service is not satisfactory.

Recommendations

Based on the results, the hotel can make the following recommendations:

The hotel should focus on improving the service quality to increase the probability of guests being satisfied.
The hotel should also consider offering additional services or amenities to increase the overall satisfaction level of guests.

Limitations of the Study

There are several limitations of this study. Firstly, the study only surveyed 100 guests, which may not be a representative sample of the entire population. Secondly, the study only considered the satisfaction level of guests with the service, and did not consider other factors that may affect the satisfaction level.

Future Research Directions

Future research directions include:

Conducting a larger survey to obtain a more representative sample of the population.
Considering other factors that may affect the satisfaction level of guests, such as the price of the service, the location of the hotel, and the quality of the amenities.

Conclusion

In conclusion, the hotel can use the probability of 0.75 to determine whether its service is satisfactory to its customers. The hotel should focus on improving the service quality to increase the probability of guests being satisfied. Future research directions include conducting a larger survey and considering other factors that may affect the satisfaction level of guests.

References

[1] Johnson, R. A. (2013). Probability and statistics for engineers. New York: McGraw-Hill.
[2] Moore, D. S. (2014). The basic practice of statistics. New York: W.H. Freeman and Company.

Appendix

The following is the R code used to calculate the probability:

# Load the necessary libraries
library(dplyr)

# Create a data frame from the table
df <- data.frame(
  Category = c("Food", "Service", "Cleanliness", "Overall"),
  Satisfied = c(80, 70, 85, 75),
  Not_Satisfied = c(20, 30, 15, 25)
)

# Calculate the probability
probability <- df$Satisfied[4] / (df$Satisfied[4] + df$Not_Satisfied[4])

# Print the probability
print(probability)

This code creates a data frame from the table and calculates the probability using the formula P(X = 1) = Number of guests who are satisfied / Total number of guests surveyed. The probability is then printed to the console.

Introduction

In our previous article, we discussed how a hotel can determine whether its service is satisfactory to its customers by surveying 100 guests and calculating the probability of a randomly selected guest being satisfied. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the purpose of surveying guests?

A: The purpose of surveying guests is to determine the satisfaction level of guests with the service provided by the hotel. This information can be used to improve the service quality and increase customer satisfaction.

Q: How do I calculate the probability of a guest being satisfied?

A: To calculate the probability of a guest being satisfied, you need to divide the number of guests who are satisfied by the total number of guests surveyed.

Q: What is the formula for calculating the probability?

A: The formula for calculating the probability is:

P(X = 1) = Number of guests who are satisfied / Total number of guests surveyed

Q: What is the significance of the probability value?

A: The probability value indicates the likelihood of a guest being satisfied with the service. A probability value greater than 0.5 indicates that the service is satisfactory, while a probability value less than 0.5 indicates that the service is not satisfactory.

Q: How can I improve the service quality?

A: To improve the service quality, you can consider the following steps:

Identify the areas where the service is not satisfactory
Develop strategies to improve the service quality in those areas
Implement the strategies and monitor the results
Continuously evaluate and improve the service quality

Q: What are some common mistakes to avoid when surveying guests?

A: Some common mistakes to avoid when surveying guests include:

Not surveying a representative sample of guests
Not using a standardized survey instrument
Not collecting data in a consistent and reliable manner
Not analyzing the data correctly

Q: How can I ensure that the survey results are accurate and reliable?

A: To ensure that the survey results are accurate and reliable, you can consider the following steps:

Use a standardized survey instrument
Collect data in a consistent and reliable manner
Analyze the data correctly
Validate the results using multiple sources

Q: What are some benefits of surveying guests?

A: Some benefits of surveying guests include:

Improving the service quality
Increasing customer satisfaction
Enhancing the reputation of the hotel
Identifying areas for improvement

Q: How often should I survey guests?

A: The frequency of surveying guests depends on the specific needs of the hotel. However, it is generally recommended to survey guests at least once a year to ensure that the service quality is maintained and improved.

Q: What are some common challenges associated with surveying guests?

A: Some common challenges associated with surveying guests include:

Low response rates
Biased responses
Difficulty in collecting data
Analyzing the data correctly

Q: How can I overcome these challenges?

A: To overcome these challenges, you can consider the following steps:

Use a standardized survey instrument
Collect data in a consistent and reliable manner
Analyze the data correctly
Validate the results using multiple sources

Conclusion

In conclusion, surveying guests is an essential step in determining whether the service is satisfactory to its customers. By understanding the purpose of surveying guests, calculating the probability of a guest being satisfied, and avoiding common mistakes, you can ensure that the survey results are accurate and reliable. By following the steps outlined in this article, you can improve the service quality, increase customer satisfaction, and enhance the reputation of the hotel.