Lesson 4: Introduction to Statistical Analysis

Objectives:

1. Understand the basics of statistical analysis.
2. Learn about hypothesis testing and confidence intervals.
3. Apply statistical methods to analyze data.

1. Basics of Statistical Analysis

Statistical analysis involves collecting, reviewing, and drawing conclusions from data. It helps in understanding data patterns, making predictions, and making informed decisions.

Key Concepts:

• Probability Distributions: Functions that describe the likelihood of different outcomes.
• Central Limit Theorem: States that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population distribution.
• Sampling Methods: Techniques to select a subset of data from a larger population.

2. Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. The two types of hypotheses are:

• Null Hypothesis (H0): Assumes no effect or no difference.
• Alternative Hypothesis (H1): Assumes an effect or a difference.

Steps in Hypothesis Testing:

1. Formulate Hypotheses: State the null and alternative hypotheses.
2. Select Significance Level (α): Common choices are 0.05 or 0.01.
3. Calculate Test Statistic: Use a statistical test (e.g., t-test) to compute the test statistic.
4. Determine p-value: Probability of obtaining results at least as extreme as the observed results.
5. Make Decision: Reject or fail to reject the null hypothesis based on the p-value and significance level.

Code Example: Hypothesis Testing with SciPy

from scipy import stats

# Sample data
data1 = [23, 25, 30, 35, 40]
data2 = [22, 27, 29, 34, 37]

# Performing t-test
t_statistic, p_value = stats.ttest_ind(data1, data2)

print(f"T-statistic: {t_statistic}")
print(f"P-value: {p_value}")

# Interpreting the result
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis.")
else:
    print("Fail to reject the null hypothesis.")

3. Confidence Intervals

A confidence interval provides a range of values that is likely to contain the population parameter with a certain level of confidence (e.g., 95%).

Code Example: Calculating Confidence Interval

import numpy as np
import scipy.stats as stats

# Sample data
data = [23, 25, 30, 35, 40]
mean = np.mean(data)
std_dev = np.std(data, ddof=1)
n = len(data)

# 95% Confidence Interval
confidence = 0.95
z_score = stats.norm.ppf((1 + confidence) / 2)
margin_of_error = z_score * (std_dev / np.sqrt(n))

confidence_interval = (mean - margin_of_error, mean + margin_of_error)
print(f"95% Confidence Interval: {confidence_interval}")

4. Applying Statistical Methods

Use statistical methods to analyze data and make informed decisions. This includes:

• Correlation Analysis: Assessing the strength and direction of relationships between variables.
• Regression Analysis: Modeling the relationship between a dependent variable and one or more independent variables.

Code Example: Simple Linear Regression

from sklearn.linear_model import LinearRegression
import numpy as np

# Sample data
X = np.array([[1], [2], [3], [4], [5]])  # Independent variable
y = np.array([1.1, 2.1, 2.9, 4.1, 5.2])  # Dependent variable

# Creating and fitting the model
model = LinearRegression()
model.fit(X, y)

# Making predictions
predictions = model.predict(X)

print(f"Intercept: {model.intercept_}")
print(f"Coefficient: {model.coef_}")
print(f"Predictions: {predictions}")

5. Key Takeaways

• Statistical analysis helps in making informed decisions based on data.
• Hypothesis testing and confidence intervals are fundamental methods in statistics.
• Applying these methods effectively requires understanding and interpreting the results accurately.

6. Homework/Practice:

• Perform hypothesis testing on a dataset to test a specific hypothesis.
• Calculate confidence intervals for sample data.
• Apply correlation and regression analysis to explore relationships between variables.