Statistical inference is a critical aspect of data analysis and machine learning. It involves making predictions or generalizations about a population based on a sample of data. This module will cover the fundamental concepts of statistical inference, including hypothesis testing, confidence intervals, and p-values.

• Population: The entire group of individuals or instances about whom we hope to learn.
• Sample: A subset of the population, selected for analysis.

• Parameter: A numerical characteristic of a population (e.g., population mean).
• Statistic: A numerical characteristic of a sample (e.g., sample mean).

Hypothesis testing is a method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis.

• Null Hypothesis (H0): A statement that there is no effect or no difference.
• Alternative Hypothesis (H1): A statement that there is an effect or a difference.
• Significance Level (α): The probability of rejecting the null hypothesis when it is true (commonly set at 0.05).

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%).

The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true.

1. State the Hypotheses:

   • Null Hypothesis (H0): μ = μ0
   • Alternative Hypothesis (H1): μ ≠ μ0

2. Choose the Significance Level (α):

   • Common choices are 0.05, 0.01, or 0.10.

3. Calculate the Test Statistic:

   • For a sample mean: \( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \)
   • Where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the population mean under H0, \( s \) is the sample standard deviation, and \( n \) is the sample size.

4. Determine the Critical Value or P-Value:

   • Compare the test statistic to a critical value from the t-distribution table, or calculate the p-value.

5. Make a Decision:

   • If the test statistic exceeds the critical value or if the p-value is less than α, reject the null hypothesis.

Suppose we want to test if the average height of students in a school is 170 cm. We take a sample of 30 students and find the sample mean height to be 172 cm with a standard deviation of 5 cm.

1. State the Hypotheses:

   • H0: μ = 170
   • H1: μ ≠ 170

2. Choose the Significance Level:

   • α = 0.05

3. Calculate the Test Statistic:

   import scipy.stats as stats
   import numpy as np
   
   sample_mean = 172
   population_mean = 170
   sample_std = 5
   sample_size = 30
   
   t_statistic = (sample_mean - population_mean) / (sample_std / np.sqrt(sample_size))
   print(f'Test Statistic: {t_statistic:.2f}')
   

4. Determine the Critical Value or P-Value:

   p_value = 2 * (1 - stats.t.cdf(np.abs(t_statistic), df=sample_size-1))
   print(f'P-Value: {p_value:.4f}')
   

5. Make a Decision:

   • If p-value < 0.05, reject H0.

• We import necessary libraries (scipy.stats and numpy).
• We define the sample mean, population mean, sample standard deviation, and sample size.
• We calculate the t-statistic using the formula.
• We calculate the p-value using the cumulative distribution function (CDF) of the t-distribution.
• We compare the p-value to the significance level to make a decision.

Exercise: Perform a hypothesis test to determine if the average weight of a certain species of fish is 50 kg. You have a sample of 25 fish with a mean weight of 52 kg and a standard deviation of 4 kg. Use a significance level of 0.05.

Solution:

1. State the Hypotheses:

   • H0: μ = 50
   • H1: μ ≠ 50

2. Choose the Significance Level:

   • α = 0.05

3. Calculate the Test Statistic:

   sample_mean = 52
   population_mean = 50
   sample_std = 4
   sample_size = 25
   
   t_statistic = (sample_mean - population_mean) / (sample_std / np.sqrt(sample_size))
   print(f'Test Statistic: {t_statistic:.2f}')
   

4. Determine the Critical Value or P-Value:

   p_value = 2 * (1 - stats.t.cdf(np.abs(t_statistic), df=sample_size-1))
   print(f'P-Value: {p_value:.4f}')
   

5. Make a Decision:

   • If p-value < 0.05, reject H0.

• Mistake: Confusing the null hypothesis with the alternative hypothesis.
  • Tip: Clearly define H0 and H1 before starting the test.
• Mistake: Using the wrong formula for the test statistic.
  • Tip: Ensure you use the correct formula based on the type of test (e.g., one-sample t-test, two-sample t-test).
• Mistake: Misinterpreting the p-value.
  • Tip: Remember that a low p-value (< α) indicates strong evidence against the null hypothesis.

In this section, we covered the basics of statistical inference, including hypothesis testing, confidence intervals, and p-values. We also walked through a practical example of a one-sample t-test and provided an exercise to reinforce the concepts. Understanding these fundamental concepts is crucial for making informed decisions based on data, which is a key aspect of machine learning.