JD Power And Associates Publishes Annual Results

Each Year Jd Power And Associates Publishes The Results Of Its No

Each year J.D. Power and Associates publishes the results of its North American Hotel Guest Satisfaction Index Study. For 2009, the study revealed that 66% of hotel guests were aware of the hotel’s “green” conservation program. Among these guests, 72% actually participated in the program by reusing towels and bed linens.

In a random sample of 15 hotel guests, consider the number (x) of guests who were aware of and participated in the hotel’s conservation efforts.

Paper For Above instruction

The study conducted by J.D. Power and Associates provides insightful data into guest awareness and participation in hotel sustainability initiatives. Analyzing this data through probability models allows us to understand the likely outcomes and assess claims made by hotel chains regarding guest participation rates. Specifically, the binomial distribution offers an effective framework for modeling the number of guests in a sample who are both aware of and participate in the conservation program. This paper discusses the rationale for using the binomial model, estimates the probability of success, calculates probabilities for specific scenarios under certain assumptions, and evaluates the claim made by a hotel chain using the normal approximation to the binomial distribution.

Part A: Justification for Using the Binomial Random Variable

The variable x represents the number of hotel guests in a sample of 15 who are aware of and participate in the conservation program. To identify if x approximates a binomial random variable, we examine the main conditions defining the binomial model:

  • Fixed number of trials: The sample consists of 15 guests, which constitutes a fixed number of independent trials.
  • Binary outcomes: Each guest either participates (success) or does not participate (failure) in the conservation program.
  • Constant probability: The probability that any guest participates remains constant within the sample.
  • Independence: The participation outcome for each guest is independent of others, assuming a random sample.

Given these conditions, the variable x, representing the count of successes, follows approximately a binomial distribution B(n=15, p), where p is the probability of any guest both being aware and participating. This approximation assumes the sample is random, and individual responses are independent, which is reasonable in this situation, making x an appropriate binomial random variable.

Part B: Estimating the Probability p

From the study results, 66% of hotel guests are aware of the conservation program, and among these, 72% participate. The overall probability that a randomly selected guest both is aware of and participates in the program can be determined by multiplying these probabilities:

p = P(guest is aware) × P(participates | aware) = 0.66 × 0.72 = 0.4752.

Thus, the estimated probability p that a guest both is aware of and participates in the conservation efforts is approximately 0.4752.

Part C: Probability of At Least 10 Out of 15 Guests Participating (Assuming p=0.45)

Given the assumption p=0.45, we utilize the binomial distribution B(n=15, p=0.45) to find the probability that at least 10 guests are aware and participate:

P(x ≥ 10) = 1 - P(x ≤ 9) = 1 - \sum_{k=0}^{9} \binom{15}{k} p^{k} (1-p)^{15-k}.

Calculations can be simplified via normal approximation, noting that n×p=6.75 and n×(1-p)=8.25, which satisfy the criteria for approximation. The mean and standard deviation of the distribution are:

  • Mean μ = n×p = 15 × 0.45 = 6.75.
  • Standard deviation σ = √(n×p×(1-p)) = √(15×0.45×0.55) ≈ √(3.7125) ≈ 1.924.

Applying the normal approximation with a continuity correction, P(x ≥ 10) ≈ P(Y ≥ 9.5), where Y ~ N(6.75, 1.924^2). Converting to z-score:

z = (9.5 - 6.75) / 1.924 ≈ 2.75 / 1.924 ≈ 1.43.

Using standard normal tables, P(Z ≥ 1.43) ≈ 0.0764. Therefore, the probability that at least 10 of the 15 guests participate is approximately 7.64%.

Part D: Testing the Hotel Chain’s Claim Using Normal Approximation

The hotel chain claims that more than 110 out of 200 guests participate in the conservation program. To evaluate this claim, we model the number of participating guests as a binomial variable B(200, p=0.4752) (based on historical data). The expected number of participants is:

  • μ = n×p = 200 × 0.4752 ≈ 95.04.

The standard deviation is:

  • σ = √(n×p×(1-p)) = √(200×0.4752×0.5248) ≈ √(49.97) ≈ 7.07.

To assess whether observing more than 110 participants is unusual under this model, we again use the normal approximation with continuity correction. We compute the z-score for 110.5:

z = (110.5 - 95.04) / 7.07 ≈ 15.46 / 7.07 ≈ 2.19.

The probability of observing more than 110 participants (z > 2.19) is approximately 0.0143, or 1.43%. Since this probability is very low, observing more than 110 participants would be considered statistically significant evidence against the historical participation rate, leading us to believe that the claim that “more than 110 guests participated” is plausible. However, given the low p-value, it could also suggest a change in guest engagement levels.

Conclusion

This analysis demonstrates the utility of probability and statistical tools in evaluating hotel guest participation in conservation programs. The binomial distribution effectively models the count of successes in small, independent samples, while normal approximation provides a practical method for calculating probabilities for larger samples. The results indicate that while the probability of high participation rates varies with p, observed outcomes like over 110 participants out of 200 are statistically noteworthy and should be interpreted within the context of potential shifts in guest engagement behaviors.

References

  • Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson.
  • Bluman, A. G. (2017). Elementary Statistics: A Step By Step Approach. McGraw-Hill Education.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and Sciences. Cengage Learning.
  • Freeman, J. (2019). The use of the normal approximation to the binomial distribution. Journal of Statistical Computation and Simulation, 89(1), 45-54.
  • Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583-621.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Peęcherz, J. (2020). Approximating binomial probabilities with the normal distribution. Statistics in Transition, 21(1), 115-123.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering & the Sciences. Pearson.
  • Zar, J. H. (2010). Biostatistical Analysis. Pearson.

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