Basic inferential statistics with examples

Here are five examples related to different steps in inferential stats:

1. **Sampling:**

   - Example 1: In a clinical trial for a new drug, a random sample of patients is selected to receive the experimental treatment, while another group receives a placebo.

   - Example 2: A hospital conducts a patient satisfaction survey by randomly sampling 500 patients from their admission records.

   - Example 3: Researchers select a random sample of medical records to study the prevalence of a specific condition in a population.

   - Example 4: A public health agency conducts a random household survey to estimate the vaccination coverage rate in a community.

   - Example 5: A clinical researcher selects a random sample of cancer patients to study the effectiveness of a particular chemotherapy regimen.

2. **Population and Sample:**

   - Example 1: The population consists of all patients with diabetes in the city, while the sample includes 200 diabetic patients from a local clinic.

   - Example 2: In a study on heart disease, the population is all adults in the country, and the sample comprises 1,000 adults randomly chosen from various regions.

   - Example 3: The population of interest is all infants born prematurely, while the sample is the group of premature infants admitted to a neonatal intensive care unit.

   - Example 4: A study aims to assess antibiotic resistance in bacteria. The population is all bacteria in a specific region, and the sample consists of bacterial cultures from various sources.

   - Example 5: Researchers investigate the prevalence of a rare genetic disorder in a country, with the population being all citizens and the sample being those diagnosed with the disorder.

3. **Central Limit Theorem:**

   - Example 1: In a study of blood pressure in a population, as the sample size increases, the distribution of sample means approaches a normal distribution.

   - Example 2: Researchers measure cholesterol levels in multiple random samples of 100 patients each and observe that the means of these samples follow a bell-shaped curve.

   - Example 3: When studying the heights of individuals in different neighborhoods, the distribution of sample means becomes approximately normal as more neighborhoods are sampled.

   - Example 4: In a clinical trial assessing pain relief medications, the average pain scores among random samples of patients tend to cluster around a central value.

   - Example 5: In a hospital, the distribution of average daily patient admissions over a year follows a normal distribution as more years are considered.

4. **Hypothesis Testing:**

   - Example 1: Researchers hypothesize that a new treatment reduces the duration of hospital stays for patients with a particular condition.

   - Example 2: A clinician tests whether a new diagnostic tool accurately identifies patients with a rare disease.

   - Example 3: A study examines whether there is a significant difference in blood glucose levels between patients following different diet plans.

   - Example 4: Researchers investigate whether a specific exercise regimen improves the quality of life for cancer survivors.

   - Example 5: A clinical trial assesses whether a drug has a statistically significant effect on lowering blood pressure in a hypertensive population.

5. **Confidence Intervals:**

   - Example 1: Researchers estimate the 95% confidence interval for the mean age of patients in a cardiology clinic to be between 55 and 65 years.

   - Example 2: A public health survey estimates the 90% confidence interval for the vaccination coverage rate in a community to be 80% to 85%.

   - Example 3: In a study on medication effectiveness, the 99% confidence interval for the reduction in pain scores is calculated as 3.5 to 5.2 points.

   - Example 4: Researchers estimate the 95% confidence interval for the odds ratio of a specific adverse event associated with a drug.

   - Example 5: A clinical trial reports the 80% confidence interval for the difference in survival rates between two treatment groups in a cancer study.

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Two examples:

Problem 1: Clinical Research Study Design

Step 1: Define the Problem

Problem: A clinical researcher wants to investigate whether a new drug reduces the risk of heart attacks in patients with a specific heart condition.

Step 2: Formulate a Hypothesis

Hypothesis: The new drug reduces the risk of heart attacks in patients with the heart condition.

Step 3: Plan the Study

Identify the target population: Patients with the specific heart condition.

Determine the sample size needed for adequate statistical power (e.g., 500 patients).

Randomly select patients from the target population.

Divide them into two groups: one receiving the new drug (treatment group) and one receiving a placebo (control group).

Step 4: Data Collection

Measure baseline characteristics of patients in both groups:

Treatment Group: Mean age = 60, Gender (M/F) = 40/60, Medical history = diverse

Control Group: Mean age = 58, Gender (M/F) = 45/55, Medical history = diverse

Follow the patients for one year, tracking heart attack occurrences:

Treatment Group: 10 heart attacks

Control Group: 15 heart attacks

Record any adverse effects and medication compliance.

Step 5: Data Analysis

Use a chi-square test to compare the incidence of heart attacks between the treatment and control groups.

Calculate the relative risk: RR = (10/500) / (15/500) = 0.67.

Determine statistical significance (e.g., p < 0.05).

Step 6: Interpret Results

Statistical analysis shows that the new drug significantly reduces the risk of heart attacks compared to the placebo.

Clinical significance: A relative risk reduction of 33% is clinically meaningful.

Limitations: Potential biases, side effects, and generalizability concerns should be discussed.

Step 7: Draw Conclusions and Make Recommendations

Recommend the use of the new drug for patients with the heart condition.

Suggest further research to explore long-term effects and side effects.

Publish findings in a medical journal.

Problem 2: Patient Data Analysis

Step 1: Define the Problem

Problem: A healthcare administrator wants to optimize the scheduling of surgeries in a hospital to reduce patient waiting times.

Step 2: Formulate a Hypothesis

Hypothesis: Adjusting surgery schedules can decrease patient waiting times.

Step 3: Data Collection

Collect historical data on surgery schedules and patient waiting times for the past year.

Data includes surgery type, surgeon, patient arrival times, and waiting times.

Step 4: Data Analysis

Analyze the data using statistical software.

Calculate average waiting times for different surgery schedules and surgeons.

Identify correlations between scheduling factors and waiting times.

Step 5: Interpret Results

Analysis reveals that surgeries scheduled in the morning have an average waiting time of 30 minutes, while afternoon surgeries have an average waiting time of 45 minutes.

Surgeon A has an average waiting time of 35 minutes, while Surgeon B has an average waiting time of 50 minutes.

Statistical tests confirm significant differences in waiting times based on scheduling factors.

Step 6: Implement Changes

Adjust surgery schedules to prioritize morning surgeries for routine cases.

Allocate additional support staff during peak surgery hours.

Ensure surgeons are aware of the changes and the importance of adhering to schedules.

Step 7: Monitor and Evaluate

Continuously monitor patient waiting times after implementing changes.

Compare post-intervention waiting times to pre-intervention times.

Find that waiting times have decreased by 15-20 minutes on average.

Step 8: Report Findings

Share the results with hospital staff and administration, highlighting the improvements in patient waiting times.

Discuss the impact on overall patient satisfaction and hospital efficiency.

These examples demonstrate how data-driven approaches can be applied to solve healthcare-related problems, using both clinical research and hospital administration scenarios.