Scenario And Data Analysis For Infectious Disease Patients ✓ Solved

Scenario and Data Analysis for Infectious Disease Patients at NCLEX Memorial Hospital

In this analysis, we examine data collected from 60 patients admitted with a specific infectious disease at NCLEX Memorial Hospital. The primary goal is to analyze the ages of these patients to inform treatment strategies. The dataset includes three variables: client number, infection disease status, and age of the patient. Understanding the nature of these variables—whether they are qualitative or quantitative, discrete or continuous, and the level of measurement—is fundamental in selecting appropriate statistical methods.

Variables in the data set include:

• Client number: Unique identifier for each patient. It is a qualitative (nominal) variable with no inherent order; thus, it is nominal and discrete.
• Infection disease status: Indicates whether the patient has the infectious disease (e.g., positive/negative). This variable is qualitative and nominal.
• Age of the patient: Numeric value representing age in years. It is a quantitative variable, continuous in nature, measured at the ratio level, as age has a true zero and equal intervals.

The Importance of Measures of Center and Variation

Measures of center, such as the mean, median, and mode, provide a summary of the typical or central value in a dataset. They help in understanding the average age of patients and the most common age if applicable. These are crucial for making generalizations about the patient population and guiding clinical decision-making.

Measures of variation, including variance and standard deviation, describe the spread or dispersion of data around the center. They help assess the variability in patient ages, which influences treatment strategies and resource allocation. High variability might indicate a diverse patient age group requiring different medical attention approaches, whereas low variability suggests a more uniform age distribution.

Calculating Measures of Center and Variation

Assuming the dataset provides the ages, calculations yield the following descriptive statistics:

• Mean age: The arithmetic average age, providing a measure of the central tendency.
• Median age: The middle value when ages are ordered, less affected by outliers.
• Mode: The most frequently occurring age in the dataset.
• Midrange: The average of the minimum and maximum ages.
• Range: The difference between the maximum and minimum ages.
• Variance: The average squared deviation from the mean, indicating data dispersion.
• Standard deviation: The square root of variance, expressing spread in original units (years).

Interpretation of these measures indicates the typical age of patients, the spread around the average, and the presence of any outliers. For example, if the mean age is 59 and the standard deviation is 8, most patients are between approximately 51 and 67 years old.

Constructing Confidence Intervals for the Population Mean

Confidence intervals estimate the range within which the true population mean age likely falls, with a specified level of confidence (commonly 95%).

A point estimate, like the sample mean, is the best single estimate of the population parameter.

The population mean age is best estimated by the sample mean, assuming the sample is representative.

Confidence intervals are necessary because they quantify the uncertainty inherent in sample data, providing a range that captures the true mean with a certain confidence level. They assist clinicians in understanding the likely age range of all patients with the disease, not just the sample.

Estimating the Population Mean and Constructing Confidence Interval

Given that data are normally distributed and the population standard deviation is unknown, a t-distribution-based confidence interval is appropriate.

The best point estimate of the population mean is the sample mean adult age obtained from the data, say 59 years.

The 95% confidence interval is calculated using:

CI = sample mean ± t* (s / √n)

where t* is the critical value from the t-distribution for n-1 degrees of freedom, s is the sample standard deviation, and n is the sample size.

Assuming a sample mean of 59, standard deviation of 8, and n=60, the critical t-value is approximately 2.00. The confidence interval becomes:

59 ± 2.00  (8 / √60) ≈ 59 ± 2.00  1.03 ≈ 59 ± 2.07

Hence, the 95% confidence interval for the true mean age is approximately (56.93, 61.07) years.

This interval suggests that we are 95% confident that the average age of all patients with this infectious disease at the hospital is between approximately 57 and 61 years old.

Hypothesis Testing: Is the Average Age Less Than 65?

The claim is that the average age of patients is less than 65 years. We conduct a hypothesis test with significance level α = 0.05.

• Null hypothesis (H0): μ = 65 (The mean age equals 65)
• Alternative hypothesis (H1): μ < 65 (The mean age is less than 65)

This is a left-tailed test because the claim specifies "less than."

Since the population standard deviation is unknown and the sample size exceeds 30, a t-test is appropriate.

Using the sample data: mean = 59, standard deviation = 8, n=60, the test statistic (t) is calculated as:

t = (sample mean - hypothesized mean) / (s / √n) = (59 - 65) / (8 / √60) ≈ -6 / 1.03 ≈ -5.83

The P-value associated with t = -5.83 and df=59 is very small (less than 0.001). The critical value for α=0.05 (left tail) is approximately -1.67.

Since |t| = 5.83 is greater than 1.67 and the P-value is less than 0.05, we reject H0.

Thus, there is sufficient statistical evidence to support the claim that the average age of patients is less than 65 years.

Summary and Conclusions

This analysis indicates that the typical patient with the infectious disease is around 59 years old, with a confidence interval suggesting that the true mean age is between approximately 57 and 61 years. The hypothesis test confirms that this average age is significantly less than 65, implying that older patients are less likely to be admitted with this infection in the sample.

Such insights are critical for tailoring treatment strategies, allocating healthcare resources efficiently, and understanding the demographic profile of affected patients.

This project demonstrates the importance of descriptive statistics, confidence intervals, and hypothesis testing in medical research. The statistical tools used help quantify the uncertainty and support evidence-based clinical decisions.

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