The Owner Of A Local Restaurant Surveyed Her Staff On Their Preference Of Uniform Color. The Results Are Displayed In The Table Below. Which Pair Of Samples Is Most Representative Of The Preference Of All The

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Introduction

In statistics, a sample is a subset of a population that is used to make inferences about the population as a whole. When conducting a survey, it's essential to select a representative sample to ensure that the results accurately reflect the preferences or opinions of the entire population. In this article, we will discuss how to determine which pair of samples is most representative of the preference of all the staff members at a local restaurant.

Understanding the Problem

The owner of a local restaurant surveyed her staff on their preference of uniform color. The results are displayed in the table below:

Uniform Color Number of Staff Members
Red 15
Blue 20
Green 10
Yellow 5
Purple 10

Defining a Representative Sample

A representative sample is a subset of the population that accurately reflects the characteristics of the population as a whole. In this case, we want to determine which pair of samples is most representative of the preference of all the staff members.

Calculating the Probability of Each Pair

To determine which pair of samples is most representative, we need to calculate the probability of each pair. We can use the concept of probability to calculate the likelihood of each pair.

Pair 1: Red and Blue

The probability of the pair Red and Blue is calculated as follows:

P(Red and Blue) = (Number of staff members who prefer Red) / (Total number of staff members) × (Number of staff members who prefer Blue) / (Total number of staff members)

P(Red and Blue) = 15/50 × 20/50 = 0.3

Pair 2: Blue and Green

The probability of the pair Blue and Green is calculated as follows:

P(Blue and Green) = (Number of staff members who prefer Blue) / (Total number of staff members) × (Number of staff members who prefer Green) / (Total number of staff members)

P(Blue and Green) = 20/50 × 10/50 = 0.4

Pair 3: Green and Yellow

The probability of the pair Green and Yellow is calculated as follows:

P(Green and Yellow) = (Number of staff members who prefer Green) / (Total number of staff members) × (Number of staff members who prefer Yellow) / (Total number of staff members)

P(Green and Yellow) = 10/50 × 5/50 = 0.1

Pair 4: Yellow and Purple

The probability of the pair Yellow and Purple is calculated as follows:

P(Yellow and Purple) = (Number of staff members who prefer Yellow) / (Total number of staff members) × (Number of staff members who prefer Purple) / (Total number of staff members)

P(Yellow and Purple) = 5/50 × 10/50 = 0.1

Pair 5: Purple and Red

The probability of the pair Purple and Red is calculated as follows:

P(Purple and Red) = (Number of staff members who prefer Purple) / (Total number of staff members) × (Number of staff members who prefer Red) / (Total number of staff members)

P(Purple and Red) = 10/50 × 15/50 = 0.3

Determining the Most Representative Pair

To determine which pair of samples is most representative, we need to compare the probabilities of each pair. The pair with the highest probability is the most representative.

Conclusion

In this article, we discussed how to determine which pair of samples is most representative of the preference of all the staff members at a local restaurant. We calculated the probability of each pair and determined that the pair Blue and Green is the most representative.

Recommendations

Based on the results, the owner of the local restaurant should consider the following recommendations:

  • Blue uniforms: The owner should consider offering blue uniforms as an option for the staff members, as it is the most preferred color.
  • Green uniforms: The owner should also consider offering green uniforms as an option, as it is the second most preferred color.
  • Staff feedback: The owner should encourage staff feedback and suggestions to ensure that the uniforms are comfortable and suitable for the staff members.

Limitations

This analysis has some limitations. The sample size is relatively small, and the results may not be generalizable to the entire population. Additionally, the analysis assumes that the staff members' preferences are independent of each other, which may not be the case in reality.

Future Research Directions

Future research directions could include:

  • Larger sample size: Conducting a survey with a larger sample size to increase the accuracy of the results.
  • More detailed analysis: Conducting a more detailed analysis of the staff members' preferences, including their age, gender, and job title.
  • Comparison with other options: Comparing the results with other options, such as different uniform colors or styles.

Q: What is the purpose of a representative sample?

A: A representative sample is a subset of the population that accurately reflects the characteristics of the population as a whole. The purpose of a representative sample is to make inferences about the population based on the sample.

Q: How is a representative sample selected?

A: A representative sample is selected using a random sampling method, such as simple random sampling or stratified random sampling. This ensures that every member of the population has an equal chance of being selected.

Q: What are the characteristics of a representative sample?

A: A representative sample has the following characteristics:

  • Random selection: The sample is selected randomly from the population.
  • Equal probability: Every member of the population has an equal chance of being selected.
  • No bias: The sample is free from bias and does not reflect any particular subgroup or characteristic of the population.

Q: How is the probability of a pair of samples calculated?

A: The probability of a pair of samples is calculated using the formula:

P(A and B) = (Number of staff members who prefer A) / (Total number of staff members) × (Number of staff members who prefer B) / (Total number of staff members)

Q: What is the most representative pair of samples?

A: The most representative pair of samples is the pair with the highest probability. In this case, the pair Blue and Green has the highest probability.

Q: What are the limitations of this analysis?

A: The limitations of this analysis include:

  • Small sample size: The sample size is relatively small, which may not be representative of the entire population.
  • Assumption of independence: The analysis assumes that the staff members' preferences are independent of each other, which may not be the case in reality.

Q: What are the recommendations for the owner of the local restaurant?

A: The recommendations for the owner of the local restaurant include:

  • Blue uniforms: Consider offering blue uniforms as an option for the staff members, as it is the most preferred color.
  • Green uniforms: Consider offering green uniforms as an option, as it is the second most preferred color.
  • Staff feedback: Encourage staff feedback and suggestions to ensure that the uniforms are comfortable and suitable for the staff members.

Q: What are the future research directions?

A: The future research directions include:

  • Larger sample size: Conducting a survey with a larger sample size to increase the accuracy of the results.
  • More detailed analysis: Conducting a more detailed analysis of the staff members' preferences, including their age, gender, and job title.
  • Comparison with other options: Comparing the results with other options, such as different uniform colors or styles.

By following these recommendations and considering the limitations and future research directions, the owner of the local restaurant can make informed decisions about the uniforms and ensure that they are comfortable and suitable for the staff members.