> ## Documentation Index > Fetch the complete documentation index at: https://ai.tharung.in/llms.txt > Use this file to discover all available pages before exploring further. # Hypothesis Testing Hypothesis testing is a statistical method used to determine whether there is enough evidence from sample data to support a claim about a population. *** # Table of Contents * Null & Alternative Hypotheses * Hypothesis Testing Workflow * One-sample t-test * Two-sample t-test * p-value * Significance Level (α) * Type I & Type II Errors * Confidence Intervals * Choosing the Correct Test * Complete Python Examples * Interview Questions * Summary *** # 1. Null & Alternative Hypotheses A **hypothesis** is an assumption about a population parameter. There are always two hypotheses. ## Null Hypothesis (H₀) The null hypothesis assumes **no effect**, **no difference**, or **no relationship**. Examples: * Average salary = ₹50,000 * New medicine is not better than old medicine. * Two algorithms have equal accuracy. Example: ```text theme={null} H₀: μ = 50 ``` where * μ = population mean *** ## Alternative Hypothesis (H₁ or Hₐ) The alternative hypothesis represents what we want to prove. Examples ```text theme={null} H₁: μ ≠ 50 ``` or ```text theme={null} H₁: μ > 50 ``` or ```text theme={null} H₁: μ < 50 ``` *** ## Types of Alternative Hypotheses ### Two-tailed Testing for any difference. ```text theme={null} H₀: μ = 50 H₁: μ ≠ 50 ``` *** ### Right-tailed Testing if value is greater. ```text theme={null} H₀: μ ≤ 50 H₁: μ > 50 ``` *** ### Left-tailed Testing if value is smaller. ```text theme={null} H₀: μ ≥ 50 H₁: μ < 50 ``` *** # Hypothesis Testing Workflow ```text theme={null} State hypotheses ↓ Collect sample ↓ Choose statistical test ↓ Calculate test statistic ↓ Compute p-value ↓ Compare with α ↓ Reject or Fail to Reject H₀ ``` *** # 2. One-Sample t-test Used when comparing **one sample mean** against a known population value. ## Example A company claims average battery life is **10 hours**. Sample: ```python theme={null} [10.5, 9.8, 10.2, 10.4, 9.9] ``` Question: Is the actual average different from 10? *** ## Formula $$ t = \\frac{\\bar{x}-\\mu}{s/\\sqrt{n}} $$ where * x̄ = sample mean * μ = population mean * s = sample standard deviation * n = sample size *** ## Python Example ```python theme={null} from scipy.stats import ttest_1samp battery = [10.5, 9.8, 10.2, 10.4, 9.9] t_stat, p_value = ttest_1samp(battery, popmean=10) print("t statistic:", t_stat) print("p-value:", p_value) ``` *** ## Interpretation If ```text theme={null} p < 0.05 ``` Reject H₀ Otherwise Fail to reject H₀. *** # 3. Two-Sample t-test Used to compare means of **two independent groups**. Example Do students using Method A score differently than students using Method B? *** Sample Method A ```python theme={null} [78, 82, 85, 88, 90] ``` Method B ```python theme={null} [72, 75, 79, 80, 81] ``` *** ## Python ```python theme={null} from scipy.stats import ttest_ind group1 = [78,82,85,88,90] group2 = [72,75,79,80,81] t_stat, p = ttest_ind(group1, group2) print(t_stat) print(p) ``` *** ## Equal Variance Assumption If variances are unequal ```python theme={null} ttest_ind(group1, group2, equal_var=False) ``` This performs **Welch's t-test**. *** # Difference Between One-Sample and Two-Sample t-test | Feature | One Sample | Two Sample | | ---------------- | --------------- | ------------------ | | Number of groups | 1 | 2 | | Compare against | Known value | Another group | | Example | Mean = 50? | Group A vs Group B | | Function | `ttest_1samp()` | `ttest_ind()` | *** # 4. p-value The **p-value** measures how likely the observed data would occur if the null hypothesis were true. It is **not** the probability that the null hypothesis is true. *** ## Interpretation | p-value | Decision | | ------- | -------------------------------- | | \< 0.01 | Very strong evidence against H₀ | | \< 0.05 | Strong evidence against H₀ | | > 0.05 | Not enough evidence to reject H₀ | *** Example ```text theme={null} α = 0.05 p = 0.03 ``` Since ```text theme={null} 0.03 < 0.05 ``` Reject H₀. *** Another Example ```text theme={null} p = 0.18 ``` Cannot reject H₀. *** # 5. Significance Level (α) The significance level is the threshold for deciding whether to reject the null hypothesis. Common choices ```text theme={null} 0.10 0.05 0.01 ``` Most commonly ```text theme={null} α = 0.05 ``` Meaning Accept a **5% chance** of incorrectly rejecting a true null hypothesis. *** Decision Rule ```text theme={null} If p ≤ α Reject H₀ ``` Otherwise ```text theme={null} Fail to Reject H₀ ``` *** # 6. Type I & Type II Errors ## Type I Error (False Positive) Rejecting a true null hypothesis. Reality: ```text theme={null} Medicine doesn't work. ``` Conclusion: ```text theme={null} Medicine works. ``` Wrong conclusion. Probability ```text theme={null} α ``` *** ## Type II Error (False Negative) Failing to reject a false null hypothesis. Reality ```text theme={null} Medicine works. ``` Conclusion ```text theme={null} Medicine doesn't work. ``` Probability ```text theme={null} β ``` *** ## Statistical Power Power ```text theme={null} Power = 1 - β ``` Higher power means a better chance of detecting a true effect. *** ## Error Table | Reality | Decision | Result | | -------- | -------------- | ------------- | | H₀ True | Reject | Type I Error | | H₀ True | Fail to Reject | Correct | | H₀ False | Reject | Correct | | H₀ False | Fail to Reject | Type II Error | *** # 7. Confidence Intervals A confidence interval gives a range where the true population parameter is likely to lie. Example ```text theme={null} 95% CI (48.2, 52.6) ``` Interpretation: Using this method repeatedly, about 95% of the intervals constructed would contain the true population mean. *** ### Formula ```text theme={null} Confidence Interval = x̄ ± t* × (s / √n) ``` Where: * `x̄` = Sample mean * `t*` = Critical t-value (depends on confidence level and degrees of freedom) * `s` = Sample standard deviation * `n` = Sample size *** ## Python Example ```python theme={null} import numpy as np from scipy import stats data = [10,11,9,10,12,11,9] confidence = 0.95 mean = np.mean(data) sem = stats.sem(data) interval = stats.t.interval( confidence, len(data)-1, loc=mean, scale=sem ) print(interval) ``` *** ## Interpretation If the confidence interval contains the hypothesized value ```text theme={null} Example 95% CI (48, 53) Hypothesized mean = 50 ``` Since ```text theme={null} 50 lies inside interval ``` Fail to reject H₀ at the 5% significance level. *** # Choosing the Correct Test | Situation | Test | | ---------------------------------- | ----------------- | | Compare one sample with known mean | One-sample t-test | | Compare two independent groups | Two-sample t-test | | Compare paired measurements | Paired t-test | | Compare more than two groups | ANOVA | *** # Complete Python Example ```python theme={null} import numpy as np from scipy.stats import ttest_1samp, ttest_ind # ---------------------------- # One Sample t-test # ---------------------------- sample = [52, 49, 51, 48, 50, 52, 53] t, p = ttest_1samp(sample, popmean=50) print("One Sample Test") print("t =", t) print("p =", p) # ---------------------------- # Two Sample t-test # ---------------------------- group1 = [75,80,82,78,77] group2 = [70,72,71,68,74] t, p = ttest_ind(group1, group2) print("\nTwo Sample Test") print("t =", t) print("p =", p) ``` *** # Interview Questions ### What is the null hypothesis? A statement that assumes there is no difference or no effect. *** ### What does a p-value represent? The probability of observing results at least as extreme as those obtained if the null hypothesis is true. *** ### Why use a t-test? To compare means when the population standard deviation is unknown. *** ### Difference between α and p-value? * **α** is chosen before the experiment (decision threshold). * **p-value** is computed from the observed sample data. *** ### What is a 95% confidence interval? An interval produced by a method that, over many repeated samples, would contain the true population parameter about 95% of the time. *** # Summary | Topic | Key Idea | | ---------------------- | ------------------------------------------------ | | Null Hypothesis | No effect or difference | | Alternative Hypothesis | Effect or difference exists | | One-Sample t-test | Compare one sample with known mean | | Two-Sample t-test | Compare two independent means | | p-value | Evidence against H₀ | | Significance Level (α) | Decision threshold | | Type I Error | False Positive | | Type II Error | False Negative | | Confidence Interval | Plausible range for the population parameter | | Power | Probability of detecting a true effect (`1 - β`) | *** # Key Takeaways * Start by defining **H₀** and **H₁**. * Choose the correct statistical test based on the data. * Compare the **p-value** with **α** to make a decision. * Understand the risks of **Type I** and **Type II** errors. * Use **confidence intervals** to estimate plausible values for the population parameter. * Report both the **test result** and the **confidence interval** for a more complete statistical interpretation.