# Data Analysis Project: Step-by-Step Explanation Below is a structured explanation of how to approach the described project. This example will use a **healthcare** context and focus on the **waiting times (in minutes) for patients in a hospital emergency department**. --- ## Task One: Context Selection and Data Simulation ### 1. Context and Variable Selection - **Context:** Healthcare (hospital emergency department) - **Variable:** Patient waiting time (continuous, measured in minutes) ### 2. Data Simulation - **Sample size:** 200 (within the recommended 100–300) - **Distribution:** Exponential distribution is realistic for waiting times (often right-skewed). **Python code example:** ```python import numpy as np np.random.seed(42) # for reproducibility sample_size = 200 # Suppose average waiting time is 30 minutes data = np.random.exponential(scale=30, size=sample_size) ``` --- ## Task Two: Descriptive Statistics Using `numpy` and `pandas`: **Python code example:** ```python import pandas as pd df = pd.DataFrame({'waiting_time': data}) mean = df['waiting_time'].mean() median = df['waiting_time'].median() std_dev = df['waiting_time'].std() minimum = df['waiting_time'].min() maximum = df['waiting_time'].max() ``` **Summary Table:** | Statistic | Value (minutes) | |-------------------|----------------| | Mean | *(calculated)* | | Median | *(calculated)* | | Standard deviation| *(calculated)* | | Minimum | *(calculated)* | | Maximum | *(calculated)* | --- ## Task Three: Graphical Presentation and Skewness ### 1. Histogram **Python code example:** ```python import matplotlib.pyplot as plt plt.hist(df['waiting_time'], bins=20, color='skyblue', edgecolor='black') plt.xlabel('Waiting Time (minutes)') plt.ylabel('Frequency') plt.title('Histogram of Patient Waiting Times') plt.show() ``` *Alt text: Histogram showing the distribution of patient waiting times in minutes.* ### 2. Shape Interpretation - **Shape:** Exponential distribution is typically **positively skewed** (right-skewed). That means most patients wait a short time, but a few wait much longer. --- ## Task Four: Probability Calculation ### 1. Stated Assumption > **Assumption:** Assume waiting times are normally distributed (for this calculation only). ### 2. Probability Calculation Example **Question:** What is the probability a patient waits more than 45 minutes? **Python code example:** ```python from scipy.stats import norm # Use sample mean and std_dev prob = 1 - norm.cdf(45, loc=mean, scale=std_dev) ``` - **Interpretation:** This gives the probability that a randomly selected patient waits more than 45 minutes. --- ## Task Five: 95% Confidence Interval for the Mean **Python code example:** ```python from scipy.stats import sem, t confidence = .95 n = len(df['waiting_time']) mean = df['waiting_time'].mean() std_err = sem(df['waiting_time']) h = std_err * t.ppf((1 + confidence) / 2, n - 1) ci_lower = mean - h ci_upper = mean + h ``` - **Interpretation:** There is a 95% chance the true average waiting time is between `ci_lower` and `ci_upper` minutes. --- ## Task Six: Hypothesis Testing ### 1. Formulate a Claim **Research Claim:** The average patient waiting time is less than 35 minutes. ### 2. State Hypotheses - **Null hypothesis (H₀):** μ ≥ 35 (average waiting time is at least 35 minutes) - **Alternative hypothesis (H₁):** μ < 35 (average waiting time is less than 35 minutes) - **Significance level:** α = .05 ### 3. Conduct the Test #### (a) P-value Method **Python code example:** ```python t_stat = (mean - 35) / (std_dev / np.sqrt(n)) p_value = t.cdf(t_stat, df=n-1) ``` - If p-value < .05, reject H₀. #### (b) Critical Value Method **Python code example:** ```python t_critical = t.ppf(.05, df=n-1) ``` - If t_stat < t_critical, reject H₀. ### 4. Compare Methods and Conclusion - **Decision:** If both methods indicate rejection of H₀, we have enough evidence to support that the mean waiting time is less than 35 minutes. - **Conclusion Example:** "Based on our analysis, there is sufficient evidence at the 5% significance level to conclude that the average patient waiting time in our emergency department is less than 35 minutes. This insight can guide resource allocation and process improvements." --- ## Summary - **Simulated data** reflects a realistic hospital scenario. - **Descriptive statistics** provide central tendency and spread. - **Histogram** helps visualize skewness. - **Probability, confidence intervals, and hypothesis testing** provide quantitative evidence for management decisions. - **All steps use Python tools as specified (numpy, pandas, matplotlib, scipy.stats).** **If you need the full Python code with sample outputs, just ask!**

# Full Python Code for Data Analysis Project Below is the complete Python code that simulates patient waiting times, computes descriptive statistics, presents data graphically, and performs inferential statistics. ## Import Libraries ```python import numpy as np import pandas as pd import matplotlib.pyplot as plt from scipy.stats import norm, sem, t ``` ## Task One: Data Simulation ### 1. Simulate Waiting Times ```python # Set random seed for reproducibility np.random.seed(42) # Sample size sample_size = 200 # Simulate waiting times (exponential distribution) data = np.random.exponential(scale=30, size=sample_size) ``` ## Task Two: Descriptive Statistics ### 1. Compute Summary Statistics ```python # Create DataFrame df = pd.DataFrame({'waiting_time': data}) # Calculate summary statistics mean = df['waiting_time'].mean() median = df['waiting_time'].median() std_dev = df['waiting_time'].std() minimum = df['waiting_time'].min() maximum = df['waiting_time'].max() # Display summary statistics summary_stats = { 'Mean': mean, 'Median': median, 'Standard Deviation': std_dev, 'Minimum': minimum, 'Maximum': maximum } print(summary_stats) ``` ## Task Three: Graphical Presentation ### 1. Draw Histogram ```python # Create a histogram plt.hist(df['waiting_time'], bins=20, color='skyblue', edgecolor='black') plt.xlabel('Waiting Time (minutes)') plt.ylabel('Frequency') plt.title('Histogram of Patient Waiting Times') plt.show() ``` ## Task Four: Probability Calculation ### 1. Calculate Probability ```python # Assume normal distribution for this calculation prob_threshold = 45 # Calculate probability that a value is greater than 45 minutes p_value = 1 - norm.cdf(prob_threshold, loc=mean, scale=std_dev) print(f"Probability of waiting time greater than {prob_threshold} minutes: {p_value:.4f}") ``` ## Task Five: Confidence Interval for Population Mean ### 1. Construct 95% Confidence Interval ```python # Confidence interval calculation confidence = 0.95 n = len(df['waiting_time']) std_err = sem(df['waiting_time']) h = std_err * t.ppf((1 + confidence) / 2, n - 1) # Confidence interval bounds ci_lower = mean - h ci_upper = mean + h print(f"95% Confidence Interval for the mean: ({ci_lower:.2f}, {ci_upper:.2f})") ``` ## Task Six: Hypothesis Testing ### 1. Formulate Hypotheses ```python # Research claim claim_mean = 35 # Null and alternative hypotheses H0 = f"μ ≥ {claim_mean}" H1 = f"μ < {claim_mean}" alpha = 0.05 # significance level ``` ### 2. Conduct Hypothesis Test #### (a) P-value Method ```python # T-statistic calculation t_stat = (mean - claim_mean) / (std_dev / np.sqrt(n)) p_value = t.cdf(t_stat, df=n-1) print(f"P-value: {p_value:.4f}") ``` #### (b) Critical Value Method ```python # Critical t-value for one-tailed test t_critical = t.ppf(alpha, df=n-1) print(f"Critical t-value: {t_critical:.4f}") ``` ### 3. Compare Methods and Conclusion ```python # Decision based on P-value method if p_value < alpha: decision = "Reject H0" else: decision = "Fail to Reject H0" # Decision based on Critical value method if t_stat < t_critical: decision_critical = "Reject H0" else: decision_critical = "Fail to Reject H0" print(f"Decision using P-value method: {decision}") print(f"Decision using Critical value method: {decision_critical}") ``` ### Final Conclusion ```python if decision == "Reject H0" and decision_critical == "Reject H0": conclusion = ("There is sufficient evidence to conclude that the average patient waiting time is " "less than 35 minutes.") else: conclusion = ("There is insufficient evidence to support that the average patient waiting time is " "less than 35 minutes.") print(conclusion) ``` --- This full code provides a comprehensive analysis, simulating patient waiting times, calculating statistics, visualizing data, and performing hypothesis testing. Adjust the parameters as needed for different scenarios or datasets.