Statistics Study Notes and Examples (PDF)

Statistics Study Notes and Examples =================================== Q1: Covers All Topics (Excluding Control Charts) ------------------------------------------------ ### 1. Descriptive Statistics Definition: Descriptive statistics summarize and describe data using measures of central tendency...

Statistics Study Notes and Examples =================================== Q1: Covers All Topics (Excluding Control Charts) ------------------------------------------------ ### 1. Descriptive Statistics Definition: Descriptive statistics summarize and describe data using measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation, range). Example: A company records the weights of 5 products as 10g, 12g, 15g, 10g, and 13g. Mean = (10 + 12 + 15 + 10 + 13) / 5 = 12g\ Median = 12g (middle value)\ Mode = 10g (most frequent) ### 2. Probability Distributions Definition: A probability distribution describes how probabilities are distributed over values of a random variable. Discrete Example: A machine has a 10% defect rate. If 3 parts are tested, find the probability exactly 1 is defective (Binomial distribution). P(X = 1) = 3 × (0.1)\^1 × (0.9)\^2 = 0.243 Continuous Example: Heights are normally distributed with mean 170 cm and standard deviation 5 cm. Probability of height between 165 and 175 cm: Use Z = (X - μ) / σ and calculate the area under the curve. ### 3. Correlation and Regression Definition: Correlation measures the linear relationship (strength/direction) between variables (-1 ≤ r ≤ 1). Regression models relationships using equations like Y = β0 + β1X. Example: Hours Studied (X) vs. Test Scores (Y): (2, 50), (4, 60), (6, 70). Correlation (r) = 1 (perfect positive).\ Regression Line: Y = 10X + 30.\ Predict score for 8 hours: Y = 10(8) + 30 = 110. ### 4. Hypothesis Testing Definition: Tests whether a claim about a population parameter is true. Example: A factory claims average production is 100 units/day. A sample of 50 days gives X̄ = 98, σ = 5, α = 0.05. Test statistic: Z = (X̄ - μ) / (σ / √n) = (98 - 100) / (5 / √50) = -2.83.\ Reject H0 (evidence production is less). ### 5. Confidence Intervals Definition: Range of values likely to contain the true population parameter. Example: Sample mean X̄ = 50, σ = 5, n = 25, α = 0.05 (95% confidence). CI = X̄ ± Z × (σ / √n) = 50 ± 1.96 × (5 / √25) = (48.04, 51.96). Q2: Descriptive Statistics and Presentation ------------------------------------------- Definition: Includes measures of central tendency, dispersion, and visualization. Example: Data: 5, 10, 15, 20, 25.\ Mean = 15, Median = 15, Range = 25 - 5 = 20.\ Visualization: Create a bar chart or histogram. Q3: Probability Distributions ----------------------------- Discrete Example: A die is rolled 10 times. Find the probability of rolling a 6 exactly 2 times. P(X = 2) = 10C2 × (1/6)\^2 × (5/6)\^8 = 0.29. Continuous Example: Battery life follows an exponential distribution with λ = 0.5.\ Probability battery lasts more than 5 hours: P(X \> 5) = e\^(-λX) = e\^(-0.5 × 5) = 0.082. Q4: Correlation and Regression ------------------------------ Example: Data: X (hours studied): 1, 2, 3; Y (scores): 50, 60, 65.\ Correlation: r = 0.98 (strong positive). Regression Line: Y = 10X + 40.\ Predict score for 4 hours: Y = 10(4) + 40 = 80. Q5: Hypothesis Testing, Confidence Intervals, and Control Charts ---------------------------------------------------------------- ### Hypothesis Testing Example A company claims defect rate p = 0.05. Sample of 100 parts shows 10 defects. Test H0: p = 0.05 vs Ha: p \> 0.05:\ Z = (p̂ - p) / √(p(1-p)/n) = (0.1 - 0.05) / √(0.05×0.95/100) = 2.05.\ Reject H0 (significant increase in defect rate). ### Control Charts Example Mean = 50, Standard deviation = 2.\ Control limits for X-bar chart (3σ):\ UCL = 50 + 3×2 = 56\ LCL = 50 - 3×2 = 44.