Hypothesis Test: Emergency Room Waiting Times At Hospital

Hypothesis Test Emergency Room Waiting Timesa Hospital Administrator

Hypothesis testing is a fundamental statistical method used to make inferences about a population parameter based on sample data. In the context of emergency room waiting times, it helps hospital administrators assess whether the observed data indicate a significant deviation from expected or specified waiting times. The scenario involves analyzing a sample of 32 patients' waiting times, with the goal of estimating the average wait and testing hypotheses about the true mean waiting time in the emergency room.

Given data include a sample mean of 196.5 minutes and a known population standard deviation of 78.4 minutes. The sample size exceeds 30, which, according to the Central Limit Theorem (CLT), justifies the use of inference methods that assume normality of the sampling distribution of the mean, even if the underlying data are not normal. This is crucial because the CLT states that the distribution of the sample mean tends toward a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the sample size is sufficiently large (n > 30) (Mendenhall, et al., 2016).

(a) Rationale for Using Normality-Based Inference Methods

Despite not knowing whether waiting times follow a normal distribution, using inference methods that rely on normality is appropriate here because the sample size is large (n=32). The CLT ensures that the sampling distribution of the mean is approximately normal, enabling the application of z-tests and confidence intervals based on the normal distribution. This approximation improves with larger samples, making it a valid approach in this scenario (Larsen & Marx, 2012).

(b) Constructing a 97% Confidence Interval for the True Mean Waiting Time

To construct the 97% confidence interval, we begin by finding the critical z-value corresponding to a 97% confidence level. Since the confidence level is 97%, the significance level α is 0.03, with α/2 = 0.015 on each tail of the standard normal distribution. Consulting the standard normal table, the z-score for 0.015 in each tail is approximately 2.17 (Laerd Statistics, 2015).

The sample mean is 196.5 minutes, and the known population standard deviation is 78.4 minutes. The standard error (SE) of the mean is calculated as:

SE = σ / √n = 78.4 / √32 ≈ 78.4 / 5.657 ≈ 13.86

The confidence interval formula is:

CI = (x̄ ± z_α/2 * SE)

Plugging in the values:

CI = (196.5 ± 2.17 * 13.86) = (196.5 ± 30.14) = (166.36, 226.64)

Thus, the 97% confidence interval for the true mean waiting time is approximately (166.36, 226.64) minutes.

(c) Interpretation of the Confidence Interval

This confidence interval means that if the process of sampling were repeated numerous times and a confidence interval was computed each time, approximately 97% of these intervals would contain the true average emergency room waiting time. Therefore, we are 97% confident that the actual mean waiting time lies between approximately 166.36 and 226.64 minutes. This range provides useful information for hospital management to understand the typical waiting experience of patients and assess whether operational adjustments are needed.

(d) Hypothesis Test to Determine if Wait Time Differs from 180 Minutes

We now test whether the average waiting time differs significantly from 180 minutes, corresponding to 3 hours. The null hypothesis (H₀) and alternative hypothesis (H_a) are set as:

• H₀: μ = 180 minutes
• H_a: μ ≠ 180 minutes

This is a two-tailed test at the 3% significance level (α=0.03). The critical z-value, as established, is ±2.17. We calculate the test statistic (z) using:

Z = (x̄ - μ₀) / (σ / √n) = (196.5 - 180) / 13.86 ≈ 16.5 / 13.86 ≈ 1.19

Next, the P-value associated with Z=1.19 is determined. Since this is a two-tailed test, the P-value is twice the area to the right of Z=1.19 in the standard normal distribution:

P( |Z| > 1.19 ) = 2 P( Z > 1.19 ) ≈ 2 0.1170 = 0.234

Because the P-value (0.234) exceeds the significance level (0.03), we fail to reject the null hypothesis. Alternatively, the test statistic |Z|=1.19 is less than the critical value of 2.17, which reinforces the conclusion that there is not enough evidence to declare a significant difference from 180 minutes.

(e) Interpretation of the P-Value

The P-value of approximately 0.234 indicates that, assuming the null hypothesis is true—that the true mean waiting time is indeed 180 minutes—there is a 23.4% probability of observing a sample mean as extreme or more extreme than 196.5 minutes purely by random chance. Since this probability is quite high, it suggests that the observed difference is not statistically significant, and we do not have sufficient evidence to conclude that the mean wait time differs from three hours. The high P-value underscores the importance of considering both statistical significance and practical implications when evaluating operational concerns in emergency departments (Fisher, 2017).

Conclusion

In summary, the application of the Central Limit Theorem justifies the use of normal approximation methods for inference regarding emergency room waiting times, given the sample size exceeds 30. The constructed 97% confidence interval suggests that the true mean waiting time likely falls within approximately 166 to 227 minutes, indicating a potentially lengthy wait relative to ideal standards. The hypothesis test conducted further supports that there is no statistically significant evidence to assert that the average wait differs from 180 minutes at the 3% significance level. These insights are crucial for hospital administrators aiming to optimize patient flow and reduce wait times, and highlight the importance of robust statistical analysis in healthcare operational management.

References

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