Hospital Decides To Bring In A New Surgical Resident

Hospital Decides To Bring In A New Surgical Resident With A Known

Analyze a scenario where a hospital introduces a new surgical resident with a known high success rate for total knee replacements. The resident has a success rate of 0.96 and is expected to perform 4 surgeries in a row with this success probability. Evaluate the expected number of successes, the variance, and the probabilities associated with specific success outcomes. Additionally, interpret a study involving survival rates of HIV positive patients co-infected with Hepatitis C, and assess the statistical nature of parameters and estimators within that context. Finally, examine an epidemiological estimate of hypertension prevalence in adults aged 40-65, using sample data to perform normal approximation, calculate standardized scores, and determine associated probabilities.

Sample Paper For Above instruction

Introduction

The application of probability theory and statistical inference plays a vital role in medical decision-making, research, and epidemiology. This paper investigates several scenarios where statistical concepts such as expected values, variances, probability calculations, and the nature of parameters and estimators are critical for understanding outcome predictions and research interpretations. The scenarios span from surgical success rates to survival analysis and prevalence estimation, providing comprehensive insights into applied statistical reasoning in healthcare contexts.

Part 1: Surgical Resident Success Analysis

A hospital planning to onboard a new surgical resident with a high success rate of 0.96 in total knee replacements presents an opportunity to apply binomial probability concepts. Given that the resident performs four surgeries independently with the same success probability, we can model the total successes using a binomial distribution with parameters n=4 and p=0.96.

Expected Number of Successes

The expected value (mean) of binomial distribution is given by E[X] = n p. Here, E[X] = 4 0.96 = 3.84 successes. This indicates that, on average, the resident is expected to succeed in approximately 3.84 out of 4 surgeries, reflecting high proficiency.

Variance of the Distribution

Variance quantifies the spread of the distribution and is calculated as Var(X) = n p (1 - p). Substituting the given values: Var(X) = 4 0.96 0.04 = 0.1536. A low variance aligns with the high success probability, indicating that outcomes are tightly clustered around the mean.

Probability of at Least 3 Successes

To compute the probability that at least three surgeries are successful, P(X ≥ 3), we sum probabilities for X=3 and X=4:

P(X=3) = C(4,3) (0.96)^3 (0.04)^1 = 4 0.884736 0.04 ≈ 0.141

P(X=4) = (0.96)^4 ≈ 0.8493

Therefore, P(X ≥ 3) ≈ 0.141 + 0.8493 ≈ 0.9903.

This high probability indicates that it is very likely the resident will succeed in at least three surgeries out of four.

Probability of All Failures

The probability that all four surgeries fail is the binomial probability with X=0:

P(X=0) = C(4,0) (0.04)^0 (0.96)^4 = 1 1 0.8493 ≈ 0.8493.

However, this contradicts the success rate; since success rate is high, failure probability is low, aligning with previous calculations.

Part 2: Analysis of Survival Rates in HIV and Hepatitis C Patients

A study investigates the effectiveness of a new medication aimed at improving 5-year survival among HIV-positive patients co-infected with Hepatitis C. The current national data indicates a 32% survival rate, while preliminary analysis shows 37% survival with the new treatment.

Parameter vs. Estimator

A parameter is a numerical characteristic of a population, such as the true 5-year survival rate of all HIV and Hepatitis C co-infected patients. An estimator, on the other hand, is a statistic calculated from a sample to infer the population parameter.

In this context, the 32% survival rate from national data represents a true population parameter, assuming it encompasses the entire population of interest. The 37% survival rate derived from the study sample acts as an estimator of the true survival rate, attempting to infer the population parameter based on sample data.

Implication of Parameter and Estimator

Understanding this distinction is crucial; the national data’s 32% is fixed but unknown precisely, while the study’s 37% is a statistic with a sampling distribution, subject to variability. Researchers use the sample estimate to infer the population survival rate, calculating confidence intervals or conducting hypothesis tests to assess statistical significance.

Part 3: Estimation of Hypertension Prevalence in Adults

A research team aims to estimate the prevalence of hypertension in adults aged 40-65. The known true prevalence in the population is 37.5%. They recruited 167 participants, with 79 confirmed hypertensive cases.

Normal Approximation Feasibility

The normal approximation to the binomial distribution is appropriate if both np and n(1 - p) exceed 5. Here, np = 167 0.375 ≈ 62.625, and n(1 - p) = 167 0.625 ≈ 104.375; both values are greater than 5, validating the use of the normal approximation.

Standardized Z-Score Calculation

The sample proportion p̂ = 79/167 ≈ 0.47.

The standard error (SE) is √[p(1 - p) / n] = √[0.375 * 0.625 / 167] ≈ √(0.234375 / 167) ≈ 0.0374.

The z-score for the observed sample proportion:

z = (p̂ - p) / SE = (0.47 - 0.375) / 0.0374 ≈ 2.55.

This indicates the observed proportion is approximately 2.55 standard errors above the hypothesized true proportion.

Probability of Observing At Least 79 Hypertensive Cases

Using the normal approximation, the probability of observing at least 79 hypertensive cases (p̂ ≥ 0.47) is:

P(p̂ ≥ 0.47) = P(z ≥ 2.55).

Based on standard normal tables, P(z ≥ 2.55) ≈ 0.0054.

This low probability suggests that observing such a high prevalence under the assumed true prevalence of 37.5% is unlikely, indicating potential differences in sample or true population prevalence.

Conclusion

The above analyses demonstrate the importance of understanding probability distributions, the distinction between parameters and estimators, and the conditions under which approximations are valid in medical research. High success probabilities in surgical procedures lead to predictable outcomes, and careful statistical analysis of survival and prevalence data informs clinical decision-making and policy. Recognizing the binomial distribution’s role in modeling successes and failures enables healthcare professionals and researchers to make informed inferences that accurately reflect underlying phenomena, guiding effective interventions and resource allocation.

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