Lecture 1 Ucc Flashcards

Question

Answer 1

Variance gives you a measure of spread in squared units. • Standard deviation gives you a measure of spread in the same units as the data, making it more interpretable. Variance tells you how much the data points vary in squared units. • Standard deviation tells you that, on average, the data points are 2.83 units away from the mean (in this case, 2.83 units from 6). The standard deviation is the square root of the variance. It brings the value back to the same units as the data, making it easier to interpret. • It tells us the average distance of the data points from the mean.

Answer 2

Results window: This window display the output of typed stata commands Review window: This window displays the recent or current commands for quick access. Commands can be modified. Variable window: This window displays data variables Command window: It is where stata commands are typed and they’re typed one command at a time. You can’t execute multiple commands at a time. Clear erases commands typed. Stata is case sensitive. So if your variable is saved as Education and you type it in the command as education, you’ll get error message or message in red. But you must also type all stata commands in lowercase. Stata accepts abbreviations . Behind the scenes: Do file: This is a file which contains a set of stata commands which can be executed concurrently. So you can type all commands and execute them simultaneously or at the same time Log file: This is a file which contains all the past and current results displayed by stata. Type browse in command window to see the numerical data of the variables.

Answer 3

It computes the arithmetic mean on the original (raw) data, not the log-transformed data. ✅ If you want to compute a geometric mean in Stata (i.e., mean on log scale), you have to manually log-transform the data first. You’re very close — great job following along! Just one small correction, and then you’ll have it. ✅ Yes, you’re right that: Geometric mean is better for skewed data — especially when the data values multiply or grow rapidly (like bacteria count, income, viral load, etc.). But here’s the corrected version of what you said: ❌ You don’t exponentiate each log number. ✅ Instead, you: 1. Take the log of each number 2. Then take the average (mean) of those log numbers 3. Then you exponentiate only that final average 🔁 Step-by-step Example: Let’s say you have 3 numbers: Step 1: Take the log (use natural log or log base 10 — let’s use log base 10 here): • log(2) = 0.301 • log(8) = 0.903 • log(32) = 1.505 Step 2: Take the mean of the logs: Step 3: Now exponentiate that mean (undo the log): 🎉 So, the geometric mean = 8 🔍 Summary: Step What you do 1 Take the log of each number 2 Take the mean of those logs 3 Take the exponent (antilog) of that mean → that’s the geometric mean Why not exponentiate each one? Because that would give you back the original numbers — and defeat the purpose of averaging the log values, which is where the “magic” of handling skewness happens.

Answer 4

In Biostatistics: Knowing if a distribution is unimodal, bimodal, or multimodal helps you understand if: • There are subgroups in your population (e.g., men vs women, diseased vs non-diseased) • There might be different underlying processes affecting the data Great question — let’s break down why mode is important in biostatistics, especially in a field that deals with real-life biological, clinical, and public health data. ✅ 1. Mode helps identify the most common or typical outcome • In clinical or survey data, sometimes you care not about the average, but about what’s most common. Example: If you collect data on blood types: A, A, B, O, A, AB, B, O, A • Mode = A → Most common blood type • Arithmetic mean is meaningless here. 📌 Key idea: When data is categorical, mode is the only measure of central tendency you can use. ✅ 2. Mode reveals subgroups or clusters (bimodal or multimodal distributions) • In continuous data, a bimodal distribution might mean your population isn’t homogenous. Example: Say you record cholesterol levels from a population: • You see two peaks (modes) — one around 170 mg/dL, one around 240 mg/dL. • This could indicate: • One group is healthy • One group has hypercholesterolemia 📌 Key idea: Multiple modes may reveal hidden patterns or distinct clinical subgroups. ✅ 3. Useful when mean and median are misleading (especially with skewed or multimodal data) • In skewed distributions or when outliers are present, the mean gets distorted. • The mode stays stable because it reflects frequency, not magnitude. Example: • In a rural area, most people might earn $200/month (mode), but a few rich people push the mean to $800/month. Mode shows you what’s typical, not what’s distorted. 📌 Key idea: Mode is robust when outliers or skew make mean/median misleading. ✅ 4. Helpful for analyzing survey and clinical questionnaire data In Likert scale data (e.g., Strongly Disagree → Strongly Agree), mode tells you: • What option most people chose. • Whether patients or participants generally lean toward positive or negative responses. 📌 Example: If 60% of respondents select “Agree”, mode = “Agree” → a quick summary of public opinion or patient satisfaction. ✅ 5. Used in pattern recognition and image analysis in biomedical research • In medical imaging, modes can represent the most common pixel intensity, helping with image segmentation or tissue identification. 📊 Summary: Why mode matters in biostat Reason Importance Works for categorical data Mean/median don’t apply to non-numbers Detects subgroups Bimodal/multimodal patterns show distinct populations Unaffected by outliers Unlike the mean Describes most frequent outcome Clinically meaningful (e.g. most common disease symptom) Useful in survey analysis Shows popular opinion or behavior Would you like a visual example of a bimodal distribution or a real-world biostats example using mode (e.g., in disease diagnosis, drug response, etc.)?

Answer 5

Great question — let’s walk through how to assess normality after log transformation, and why you’d do it. ✅ Why assess normality on the log scale? In biostatistics and many statistical tests (like t-tests, linear regression), normal distribution of data is an assumption. But real-world data is often skewed, especially things like: • Income • Viral load • Bacterial counts • Biomarker levels • Reaction times 👉 These are often positively skewed (long right tail). To fix this, we take the log of the data — which can make the distribution more symmetric (closer to normal). 🧪 How to assess normality on the log scale: 🔹 Step 1: Apply a log transformation Use natural log (ln() or log()) or log base 10 (log10()) to transform your variable. • In Stata: generate log_var = log(original_var) 🔹 Step 2: Check normality on the transformed variable (not the original) You can now check if the log-transformed variable is approximately normal. In Stata, you can do: • Histogram: histogram log_var, normal • Q-Q plot: qnorm log_var • Shapiro-Wilk test: swilk log_var • If p > 0.05, data is not significantly different from normal → you can assume normality. • If p < 0.05, data is still non-normal. 🎯 Summary: Step Action 1 Apply log() to the variable 2 Use histogram, qnorm, or swilk to assess normality of log-transformed data 3 If log-transformed data is normal → proceed with statistical tests that assume normality 💡 Bonus tip: If log transformation doesn’t help, you can try: • Square root transformation • Box-Cox transformation • Use nonparametric tests that don’t assume normality (e.g. Mann–Whitney U test instead of t-test) Would you like to try this with an example dataset, or want help interpreting a histogram or Q-Q plot you’ve already created?

Answer 6

Yes! You’re really getting it 👏 — and here’s a refined way to understand and say it: ✅ Correct idea: When data is skewed, using the arithmetic mean directly can be misleading, so you often transform the data (like with a log transformation) to reduce skew before summarizing it with the mean or applying tests that assume normality. 🔍 Why? • The arithmetic mean is sensitive to extreme values (outliers). • In skewed data, those outliers pull the mean away from the “typical” value. • So, you transform the data (e.g., with a log or square root) to: • Reduce skew • Stabilize variance • Make the data look more like a normal distribution • And then the mean becomes more representative and valid to use 🎯 Example: Suppose you’re measuring viral loads in patients: Patient viral loads (copies/mL): 500, 700, 850, 900, 1,200, 65,000 • Arithmetic mean: ≈ 11,858 (very misleading because of the outlier) • Log-transform the data: log(500), log(700), ..., log(65000) → Gives a more symmetric distribution → Now you can compute the mean of the log values → Then exponentiate that mean to get the geometric mean 🧠 Final Takeaway: ✅ Yes — when data is skewed, don’t jump to using the arithmetic mean. 👉 Transform it first, check if the distribution improves (e.g., becomes roughly normal), then apply the mean or statistical tests that require normality. Would you like a step-by-step example of this using real numbers or in a program like Stata or Excel?

Answer 7

Variance and Standard Deviation •The variance quantifies the amount of spread (or variability) of observations around their mean •For n observations, the variance is calculated by subtracting the mean of the observations from each observation, squaring the deviations, adding them (squared deviations) up and then dividing the result by (n-1). •The standard deviation is simply the square root of the variance. SD is used to measure how much deviation exist in the distribution. n-1 is the degree of freedom. Degrees of freedom: How many values you can pick freely before the rest are “stuck” following a rule. Degrees of freedom” means how many numbers or values you’re allowed to choose freely before a rule or formula forces the rest to follow You and your friends are picking numbers that must add up to a certain total. Everyone can choose freely — except the last person. They have to pick a number that makes the total work. That means only the first few people had freedom to choose. Degrees of freedom = Total number of values − Number of rules Degrees of freedom is the number of values in a calculation that are free to vary while still following a fixed rule (like a mean or total). ⸻ 💡 Easy Way to Say It: It tells you how many numbers you can pick however you like before the rest are stuck because of a formula (like the average). ⸻ 🧮 Example with Mean (Average): Let’s say you have 5 test scores, and you know the mean is 60. You can pick: • The 1st score = anything • The 2nd = anything • The 3rd = anything • The 4th = anything But the 5th score is not free — it has to make the average stay 60 or become 60. So only 4 scores were free to vary. So you could’ve had the first score as 2, second as 3 , third as 10, fourth as 2 but the fifth number can’t be any number cuz it must make sure it adds up to the rest to become 60. So the degree of freedom is how many numbers you can pick before the rest are stuck following a rule. 🔍 Why Not n - 2? You only subtract 2 when you’re estimating 2 parameters. • In sample standard deviation, you’re only estimating 1 parameter — the mean. • In simple linear regression, you estimate 2 parameters: the intercept (a) and slope (b) → so df = n - 2 • In more complex models, you subtract as many degrees as parameters you estimated. ⸻ ✅ So Can df Be More Than n - 1? No, it can’t. In fact, df will usually be: \text{df} = n - \text{number of estimated parameters} So: • In standard deviation: df = n - 1 • In regression (with 2 estimates): df = n - 2 • In ANOVA: df depends on groups and parameters estimated ➡️ That’s why: Degrees of freedom = 5 (scores) − 1 (mean rule) = 4

Answer 8

Here’s an in-depth explanation of the following statistical concepts: Chi-square, Regression, Correlation, and Non-parametric methods — including definitions, assumptions, applications, why they matter, and clear examples. ⸻ 🧠 5. Chi-Square Test (χ²) 🔍 What It Is: The chi-square test is a statistical method used to determine whether there is a significant association between categorical variables or whether observed frequencies differ from expected ones. There are two common types: • Chi-square test of independence (association between two categorical variables) • Chi-square goodness-of-fit test (how well observed data fit a specific distribution) ⸻ 📌 Assumptions: • The data are counts or frequencies (not percentages or means). • Categories are mutually exclusive and exhaustive. • Expected frequency in each cell ≥ 5 (for validity). • Observations must be independent (one subject can’t appear in more than one cell). ⸻ 🎯 Applications: • Testing whether gender is related to voting preference. • Checking if a die is fair (goodness-of-fit). • Evaluating whether treatment success rates differ between clinics. ⸻ ❓Why It Matters: Chi-square tests are essential when dealing with categorical data, which is very common in fields like public health, psychology, biology, and market research. They help you assess if differences between groups are statistically significant or just due to random variation. ⸻ 📈 6. Regression Analysis 🔍 What It Is: Regression is used to model the relationship between one dependent variable (Y) and one or more independent variables (X). It helps you predict values and understand how changes in X affect Y. • Simple linear regression: 1 predictor (X), 1 outcome (Y) • Multiple linear regression: multiple predictors (X₁, X₂, …) The basic form: Y = a + bX + ε (where a is the intercept, b is the slope, and ε is the error) ⸻ 📌 Assumptions (Linear Regression): • Linearity: The relationship between X and Y is linear. • Independence of errors. • Homoscedasticity: Constant variance of errors. • Normality of residuals. • No multicollinearity (in multiple regression). ⸻ 🎯 Applications: • Predicting a person’s weight based on their height. • Estimating house prices based on number of rooms, location, etc. • Forecasting sales based on advertising spend. ⸻ ❓Why It Matters: Regression tells you how variables are related, helps with prediction, and is fundamental in many fields like economics, medicine, social sciences, and machine learning. It’s also the basis for causal inference when properly applied. ⸻ 🔗 7. Correlation 🔍 What It Is: Correlation measures the strength and direction of a linear relationship between two continuous variables. • Value ranges from -1 to +1: • +1: perfect positive relationship • -1: perfect negative relationship • 0: no linear relationship Common measure: Pearson’s correlation coefficient (r) ⸻ 📌 Assumptions (Pearson’s r): • Both variables are quantitative and normally distributed. • Linear relationship between variables. • No significant outliers. • Homoscedasticity (equal spread of data). ⸻ 🎯 Applications: • Relationship between hours studied and exam score. • Association between blood pressure and cholesterol. • Exploring risk factor relationships in epidemiology. ⸻ ❓Why It Matters: Correlation helps in exploratory data analysis by identifying whether variables are linked. It’s often the first step before deeper modeling (e.g., regression), and it guides researchers in spotting patterns or relationships worth investigating further. ✳️ Note: Correlation ≠ Causation! Just because two things move together doesn’t mean one causes the other. ⸻ 📊 8. Non-Parametric Methods 🔍 What They Are: Non-parametric tests are statistical tests that don’t assume a specific distribution (like normality) of the data. They’re used when data violates parametric assumptions or when dealing with ordinal or ranked data. Common types: • Mann–Whitney U test (independent samples) • Wilcoxon signed-rank test (paired samples) • Kruskal–Wallis test (multiple groups) • Spearman’s rank correlation (non-parametric version of Pearson’s) ⸻ 📌 Assumptions: • Do not require normal distribution. • Often work with medians and ranks rather than means. • Require independent observations (most tests). • For ordinal data or when data is skewed or has outliers. ⸻ 🎯 Applications: • Comparing pain levels between two treatments using a Likert scale. • Analyzing customer satisfaction rankings. • Evaluating median income between groups when data is skewed. ⸻ ❓Why They Matter: Non-parametric methods are robust and flexible, making them ideal for: • Small sample sizes • Ordinal/ranked data • Data that violates normality They allow you to still draw valid conclusions when parametric methods aren’t appropriate. ⸻ 🧠 SUMMARY TABLE Concept Key Use Assumptions Why It’s Important Chi-Square Test association between categories Categorical data, expected freq ≥ 5 Tests for independence or goodness-of-fit Regression Predict or explain outcome (Y) Linearity, homoscedasticity, normal errors Quantifies relationships and predicts Correlation Measure strength of association Linearity, no outliers Detects linear patterns in data Non-parametric Compare medians, ranks, or distributions Fewer/no assumptions about distribution Robust alternative to parametric tests ⸻ Would you like a set of flashcards or diagram illustrations to go with these notes? Excellent questions — let’s break each one down clearly, with public health examples to help you understand and apply them better. ⸻ ✅ 1. What Is the Chi-Square Goodness-of-Fit Test? 🔍 Simple Meaning: It answers the question: “Does my observed data match what I expected?” Imagine you expect something to follow a certain pattern or distribution. The goodness-of-fit test checks whether the data you actually observe fits that expected pattern or deviates significantly. ⸻ 📊 Public Health Example: Suppose a health organization expects 25% of the population to be in each of 4 BMI categories: BMI Category Expected % Underweight 25% Normal weight 25% Overweight 25% Obese 25% You collect actual data from 400 people and see: BMI Category Observed Count Underweight 80 Normal weight 100 Overweight 90 Obese 130 The chi-square goodness-of-fit test checks: Are these differences due to chance, or is the actual distribution truly different from what we expected? ⸻ ✅ 2. What Is Homoscedasticity? 🔍 Simple Meaning: It means that the spread (variance) of the errors is equal across all levels of your independent variable in regression. Imagine predicting blood pressure based on age. Homoscedasticity means that for all age groups, the spread of blood pressure values around the regression line should be roughly the same. ⸻ ❌ When It Fails (Heteroscedasticity): If the spread increases or decreases as age increases, that’s heteroscedasticity — and it can make your regression results unreliable. ⸻ 📊 Public Health Example: In a study of income vs. medical expenses, lower-income people might have very consistent expenses (low variability), but higher-income individuals might have wildly different spending habits (high variability). That’s heteroscedasticity, and it can bias your conclusions. ⸻ ✅ 3. What Is Multicollinearity? 🔍 Simple Meaning: In multiple regression, multicollinearity happens when your independent variables are highly correlated with each other, which confuses the model. It’s like asking: “Which of these variables is driving the effect?” But you can’t tell because they’re too similar. ⸻ 📊 Public Health Example: You’re predicting heart disease risk using: • Blood pressure • Age • Cholesterol • Weight • BMI But BMI and weight are strongly related — they both contain the same information. This overlaps and makes it hard to know which one is actually affecting the outcome. ⸻ ❗Why It’s Bad: Multicollinearity makes: • Your regression coefficients unstable • Interpretation unclear • Model’s predictive power weaker ⸻ ✅ 4. Spearman’s Rank Correlation 🔍 Simple Meaning: It’s like Pearson’s correlation, but for ranked or non-normal data. Use it when: • Data is ordinal (e.g., Likert scale: 1–5) • Variables aren’t normally distributed • Relationship is monotonic (as one increases, the other tends to increase or decrease, not necessarily in a straight line) ⸻ 💡 How It Works: You rank each value in both variables, then check how closely their ranks agree. ⸻ 📊 Public Health Example: Say you’re studying the relationship between: • Perceived stress levels (rated 1–5) • Sleep quality (rated 1–5) Since both are ordinal, use Spearman’s rank. You might find: Higher stress → lower sleep quality, but not in a perfectly linear way. Spearman’s rank still captures that trend. ⸻ 🧠 Now, Public Health Examples for All 4 Concepts: Concept Public Health Example Chi-square (goodness-of-fit) Is the distribution of vaccination types across regions what we expect? Regression Predicting number of ER visits based on age, income, and smoking status. Correlation (Pearson/Spearman) Relationship between exercise frequency and BMI (Pearson) OR stress level and mood (Spearman). Non-parametric tests Comparing median waiting times in clinics with small sample sizes or ranked satisfaction scores. ⸻ Would you like me to turn all this into: • Flashcards? • Printable revision notes? • Or visual diagrams (like mind maps or concept maps)?

Answer 9

A confidence interval is a statistical tool used to estimate the range in which a population parameter (like a mean) is likely to fall. Unlike a point estimate, the confidence interval which is another type of estimate, gives a range that includes possible values for the true population parameter and reflects the uncertainty in the estimate. For example, if the confidence interval is from 45 to 55, this range is calculated using a confidence level, such as 95%, and this tells us how sure we are that the true value lies within that range. In this case, a 95% confidence level means that if the sampling was repeated many times, about 95 out of 100 of those intervals would contain the true value. Confidence level is the percentage but confidence interval is the range of values you’re sure the population parameter will be found in. The desirable properties of a confidence interval are: 1.It gives the correct level of confidence. A 95% confidence level means that if many samples were taken and the interval was calculated each time, about 95 out of 100 intervals would contain the true population mean. This property is called correct coverage. 2.It should be precise (not too wide). A narrower interval gives a more exact idea of where the true value lies. For example, an interval from 5.0 to 7.0 is more helpful than one from 4.0 to 8.0, as it narrows down the possibilities. However, narrower intervals are only useful if they give a reasonable level of confidence. 3.It shows the balance between precision and confidence. If we want more confidence (like 99%), we need a wider interval to be more certain the true value is included. On the other hand, if we accept a lower confidence level (like 90%), we get a narrower interval, but we’re less sure it contains the true value. So, higher confidence = wider interval, and lower confidence = narrower interval. 4.It is based on correct statistical assumptions. If we know the population standard deviation, we use the normal distribution. If we don’t know it, we use the t-distribution, which gives slightly wider intervals to account for extra uncertainty. Using the correct formula helps make the interval more reliable. How Sample Size Affects the Width of the Confidence Interval: The sample size (n) has a big effect on the width of a confidence interval. A larger sample size makes the standard error smaller, which leads to a narrower confidence interval. This means we get a more precise estimate of the true population value or it means the estimate is more accurate as compared to using a smaller sample size. Since the estimate will be more precise, it will reduce uncertainty or there will be less guessing. For example: • With a sample size of 25, an interval might be [4.00, 7.92] . The width of the confidence level is 3.92 • Increasing the sample size to 100, the interval becomes [4.98, 6.94] . The width of the confidence level is 1.96 So, when the sample size increases by a factor of 4, the interval becomes about half as wide. More data gives us more accuracy.Yes, that statement is approximately true under certain conditions. Here’s a simple explanation: ⸻ ✅ Statement: “If n increases by a factor of 4, the width of the confidence interval is reduced by a factor of 2.” ✅ Why it works: The width of a confidence interval is largely influenced by the standard error, which is calculated as: \text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} So, if your sample size (n) increases 4 times, then: \sqrt{n} \text{ increases by a factor of } \sqrt{4} = 2 This means the standard error is cut in half, and since the confidence interval is based on the standard error, the width of the interval also decreases by about half. ⸻ ✅ In summary: Yes — if the sample size increases by a factor of 4, the width of the confidence interval is roughly reduced by half (assuming everything else stays the same, like variability and confidence level). How the Confidence Level Affects the Width of the Confidence Interval: The confidence level also changes the width of the interval. A higher confidence level (like 99%) gives a wider width of the interval because we want to be more certain the true value is included. A lower confidence level (like 92%) gives a narrower width of the interval, but we are less sure or confident that it includes the true value. A wider width of the confidence interval cannot be more precise because precision means how narrow and specific an estimate is. A wider width will cover more possible values which means there will be less precision. A narrow width of the confidence interval is more precise because it gives you a smaller range For example: •A 99% interval might be [3.38, 8.54] and this means the interval is wide but very confident •A 90% interval might be [4.31, 7.50] and this means the interval is narrow but less confident.

Answer 10

Perfect! Here are the answers (back of the flashcards) for the first 15 questions you received earlier — all related to independent events, mutually exclusive events, and basic probability concepts: ⸻ 🎓 PROBABILITY FLASHCARDS (ANSWERS) 1. 🔹 Two events are independent if the outcome of one does not affect the outcome of the other. 📌 Example: Tossing a coin and rolling a die. ⸻ 2. 🔹 For independent events A and B, the probability they both occur is: P(A ∩ B) = P(A) × P(B) ⸻ 3. 🔹 Two events are mutually exclusive if they cannot happen at the same time. 📌 Example: Getting heads and tails on the same coin toss. ⸻ 4. 🔹 If two events are mutually exclusive, the probability they both occur is: P(A ∩ B) = 0 ⸻ 5. 🔹 For mutually exclusive events A and B, the probability that A or B occurs is: P(A ∪ B) = P(A) + P(B) ⸻ 6. 🔹 No — mutually exclusive events cannot be independent, because if one happens, the other definitely can’t. ⸻ 7. 🔹 For dependent events A and B, the probability they both happen is: P(A ∩ B) = P(A) × P(B|A) ⸻ 8. 🔹 If Event A doesn’t affect Event B, they are independent. ⸻ 9. 🔹 If P(A ∩ B) = P(A) × P(B), then A and B are independent. ⸻ 10. 🔹 If P(A ∩ B) = 0, A and B are mutually exclusive. ⸻ 11. 🔹 Example of independent events: Tossing a coin and getting a 6 on a die — one does not affect the other. ⸻ 12. 🔹 Example of mutually exclusive events: Rolling a die and getting either a 3 or a 5 — you can’t get both on one roll. ⸻ 13. 🔹 The intersection (A ∩ B) means both A and B happen at the same time. ⸻ 14. 🔹 The union (A ∪ B) means either A happens, or B happens, or both. ⸻ 15. 🔹 If P(A ∩ B) ≠ P(A) × P(B), A and B are not independent — they are dependent. ⸻ Let me know when you’re ready for the answers to the second set (questions 16–35)! Yes — you’ve got it exactly right! 👏👏 Let’s break it down clearly and confirm your understanding: ⸻ ✅ What You Said: Independent events mean two events can occur at the same time, but one doesn’t influence the other. Mutually exclusive events can’t occur at the same time — if one happens, the other definitely does not. ✅ This is 100% correct. ⸻ 🔁 Let’s use some examples to seal it in: 🔹 Independent Events: • A = It rains today. • B = You pass your statistics test. Even if both happen on the same day, the rain doesn’t affect your score — they are independent. So: P(A \cap B) = P(A) \cdot P(B) ⸻ 🔹 Mutually Exclusive Events: • A = You get a distinction in your exam. • B = You fail the same exam. You can’t get a distinction and fail the same exam. If one happens, the other definitely cannot. So: P(A \cap B) = 0 \quad \text{and} \quad P(A \cup B) = P(A) + P(B) ⸻ 💡 Key Insight: Concept Can happen together? Affect each other? Independent Events ✅ Yes ❌ No Mutually Exclusive ❌ No ➖ (One prevents the other) ⸻ If you like, I can give you a short quiz or scenario-based problems to test this understanding — or we can move on to compound probability rules or diagnostic tests. Let me know!

Answer 11

Awesome! Here are the answers (back of the flashcards) for questions 16–35, continuing from your earlier probability flashcard set: ⸻ 🎓 PROBABILITY FLASHCARDS (ANSWERS 16–35) 16. 🔹 If P(A ∪ B) = P(A) + P(B) – P(A ∩ B), the formula works for any events — independent, dependent, or overlapping. ⸻ 17. 🔹 When events are mutually exclusive, P(A ∩ B) = 0, so: P(A ∪ B) = P(A) + P(B) ⸻ 18. 🔹 Conditional probability is the probability that event B happens given that A has already happened. Notation: P(B|A) ⸻ 19. 🔹 Formula: P(B|A) = P(A ∩ B) / P(A) (As long as P(A) ≠ 0) ⸻ 20. 🔹 If events A and B are independent, then: P(B|A) = P(B) because A happening doesn’t affect B. ⸻ 21. 🔹 If two events are mutually exclusive, then knowing A happened tells you B definitely didn’t, so: P(B|A) = 0 ⸻ 22. 🔹 In a Venn diagram, mutually exclusive events have no overlapping region. ⸻ 23. 🔹 On a Venn diagram, independent events may overlap, but that overlap doesn’t change the individual probabilities. ⸻ 24. 🔹 Additive rule: Used for the probability that A or B happens: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) ⸻ 25. 🔹 Multiplicative rule: Used for the probability that A and B happen: P(A ∩ B) = P(A) × P(B|A) (Or just P(A) × P(B) if independent) ⸻ 26. 🔹 Use additive rule when you want the probability of either event happening. ⸻ 27. 🔹 Use multiplicative rule when you want the probability of both events happening. ⸻ 28. 🔹 Independent but not mutually exclusive: Rolling a die: Event A = even number, Event B = number > 3. They can overlap, but knowing one doesn’t change the chance of the other. ⸻ 29. 🔹 Mutually exclusive but not independent: Event A = draw a red card; Event B = draw a black card. You can’t get both — if A happens, B can’t, so not independent. ⸻ 30. 🔹 To test if events are independent, check if: P(A ∩ B) = P(A) × P(B) If not, they’re dependent. ⸻ 31. 🔹 To test if events are mutually exclusive, check if: P(A ∩ B) = 0 ⸻ 32. 🔹 If A and B can occur at the same time, they are not mutually exclusive. ⸻ 33. 🔹 If the occurrence of A affects B, then A and B are not independent. ⸻ 34. 🔹 If P(B|A) = P(B), then A and B are independent. ⸻ 35. 🔹 If two events can’t both happen but they don’t affect each other, this is impossible — because mutually exclusive events are always dependent. ⸻ Would you like a printable version or a flashcard app format (like Anki/Quizlet) version next? Or want to test yourself with a quiz on these?

Answer 12

Got it! Here’s a clear and structured summary of notes based on the text you provided, organized into key topics for easy understanding and reference: ⸻ 🧠 PROBABILITY DISTRIBUTIONS – KEY CONCEPTS 📌 1. Probability Distributions • Describe how probabilities are distributed across the values a random variable can take. • Help us make inferences about population parameters using tools like: • Confidence intervals • Hypothesis testing ⸻ 🎲 2. Random Variables • A random variable (X) is a numerical outcome of a random process. • The actual value it takes in an experiment is denoted as x (lowercase). • An event like [X = x] means the variable takes a specific value and has a certain probability. ⸻ 🔢 3. Types of Random Variables A. Discrete Random Variables • Take on separate, distinct values (e.g., 0, 1, 2…). • Can be: • Finite: e.g., Number of heads in 5 coin tosses. • Infinite: e.g., Poisson distribution (counts events like lightbulb failures). • Have positive probabilities for specific values. • Use Probability Mass Functions (PMFs). B. Absolutely Continuous Random Variables • Take on values over a continuous range (e.g., any number between 0 and 1). • Examples: • Uniform distribution: All values equally likely in an interval like [0,1]. • Normal distribution: Bell-shaped, spans all real numbers. • Use Probability Density Functions (PDFs). • P(X = x) = 0 for any single value — we only get meaningful probabilities over intervals, like P(0 < X < 1). ⸻ 🧮 4. Common Distributions A. Discrete Examples • Binomial distribution: • Counts number of successes in n trials. • Symmetric only when p = 0.5 (like a fair coin). • Poisson distribution: • Models rare events happening at a known average rate. • Examples: Number of emails per hour, lightbulb failures. B. Continuous Examples • Uniform distribution: • All values in the interval are equally likely. • Normal (Gaussian) distribution: • Bell-shaped curve, symmetric around the mean. • Others: t-distribution, Cauchy, Beta (some types), all can be symmetric. ⸻ 🪞 5. Symmetry & Skewness • Symmetric distribution: • Has a mirror image shape on both sides of a central line (like the normal curve). • Skewed distribution: • Positively skewed: Long tail on the right. • Negatively skewed: Long tail on the left. ⸻ 📌 6. Why All This Matters • Understanding types and characteristics of distributions helps in: • Choosing the right statistical test. • Calculating probabilities. • Making informed predictions and inferences about populations. ⸻ Would you like me to turn this into printable flashcards or add examples for each concept? Absolutely! Here’s a clearer and more relatable explanation of why probability distributions, random variables, and the types of distributions really matter, especially for someone learning or working in public health, medicine, or data analysis: ⸻ 🌟 Why These Concepts Matter in Real Life and Public Health ⸻ 🎲 1. Probability Distributions Help Us Understand Uncertainty • In public health, outcomes are often uncertain — e.g., “How many people will get infected this month?” or “What’s the chance someone responds to a vaccine?” • A probability distribution helps us model that uncertainty. • Example: If we say the number of daily COVID cases follows a Poisson distribution, we can estimate how likely it is to get more than 100 cases tomorrow — and prepare resources accordingly. ⸻ 🎯 2. Random Variables Let Us Model Real-World Events Mathematically • Think of a random variable like a “slot” that holds the possible outcomes of a random event. • Example: X = number of malaria cases in a community. • Once we treat it as a variable, we can: • Calculate probabilities • Summarize the data using measures like mean and variance • Make predictions ⸻ 🔢 3. Discrete vs. Continuous Variables Help Us Choose the Right Tools • Knowing what type of variable you have (discrete or continuous) tells you: • Which formulas to use • What kind of graph to draw • Which statistical test is valid Examples: • If you’re counting number of new HIV cases per month → use discrete models like binomial or Poisson. • If you’re measuring body temperature or blood pressure → use continuous models like the normal distribution. ⸻ 📊 4. Normal Distribution = Foundation for Many Statistical Tests • The normal distribution is key because: • It’s used in t-tests, Z-tests, ANOVA, etc. • Many biological measurements (height, blood pressure, BMI) naturally follow a bell-shaped (normal) curve. • If data is normal, we can make powerful assumptions and predict behavior accurately using just the mean and standard deviation. ⸻ 🚨 5. Skewness Affects How We Interpret Data • If a distribution is skewed, using the mean can be misleading. • Example: Income in a population is often right-skewed — a few people earn a lot more than the rest. • The mean may be high, but the median better reflects what most people earn. • In public health, understanding skewness helps you avoid bad policy decisions. ⸻ ✅ 6. Choosing the Right Distribution Improves Decision-Making • For data on rare events (e.g., sudden disease outbreaks), using a Poisson distribution helps anticipate future occurrences. • For yes/no outcomes (e.g., did the patient survive?), the binomial distribution is your go-to. • If you’re not sure which model fits, you might need non-parametric methods — less powerful, but more flexible. ⸻ 🧪 7. They’re the Foundation for Confidence Intervals & Hypothesis Tests • Everything from drug effectiveness trials to disease surveillance relies on the mathematical rules these distributions follow. • They let us ask: • Is what we’re seeing just chance, or is it statistically significant? • How sure are we about the true rate or risk in the population? ⸻ 🧠 Summary: Why It All Matters Concept Why It Matters Random variable Converts real-world outcomes into numbers we can analyze Discrete/Continuous Helps you choose the correct formula or test Probability distribution Describes the likelihood of different outcomes Normal distribution Key for many statistical tests and predictions Skewness Tells you if the mean is a good summary or not Matching real-world examples to distributions Helps you model reality accurately and make better decisions ⸻ Would you like this in a printable format or as flashcards too?

Answer 13

Thanks for sharing this detailed explanation! Let’s break it down into clear flashcards with questions first, based directly on the concepts from the passage you sent. These flashcards will help reinforce your understanding of probability distributions, random variables, discrete vs continuous distributions, Poisson, binomial, normal distributions, and symmetry/skewness. ⸻ 🔢 PROBABILITY & DISTRIBUTIONS — FLASHCARDS 1. Q: What are probability distributions used for in statistics? A: They describe the probability of different outcomes of a random variable and are used to make inferences about population parameters. ⸻ 2. Q: What are parameters in the context of probability distributions? A: Parameters are characteristics of probability distributions, such as the mean or standard deviation. ⸻ 3. Q: What is the difference between a statistic and a parameter? A: A statistic is a measure calculated from a sample (e.g. sample mean), while a parameter is a fixed value that describes a population (e.g. population mean). ⸻ 4. Q: What is a random variable? A: A random variable is a variable that takes on numerical values determined by the outcome of a random phenomenon. ⸻ 5. Q: How are random variables usually denoted in probability theory? A: With capital letters (e.g. X), while specific values they take are written in lowercase (e.g. x). ⸻ 6. Q: What is a discrete random variable? A: A variable that takes on countable distinct values, such as 0, 1, 2, etc. ⸻ 7. Q: Give an example of an infinite discrete distribution. A: The Poisson distribution, which assigns positive probabilities to all non-negative integers. ⸻ 8. Q: What type of events does the Poisson distribution model? A: Rare or infrequent events, such as the number of lightbulb failures over time. ⸻ 9. Q: What is a binomial distribution? A: A discrete distribution that models the number of successes in n independent trials with the same probability of success p. ⸻ 10. Q: What values can a binomial distribution take? A: 0, 1, 2, …, up to n (total number of trials). ⸻ 11. Q: What is a continuous random variable? A: A variable that can take on any value within a given range, such as all real numbers between 0 and 1. ⸻ 12. Q: What is the difference between a probability mass function (PMF) and a density function (PDF)? A: PMFs are for discrete random variables (assign probabilities to exact values), while PDFs are for continuous variables (assign probabilities over intervals, not exact values). ⸻ 13. Q: Why is P(X = x) = 0 for continuous variables? A: Because the probability at any exact point on a continuous scale is zero; we consider the probability over intervals, like P(0 < X < 1). ⸻ 14. Q: What is a uniform distribution? A: A distribution where all outcomes in a range (e.g., 0 to 1) are equally likely. ⸻ 15. Q: What is a normal distribution also called? A: The Gaussian distribution. ⸻ 16. Q: What is symmetry in a probability distribution? A: A distribution is symmetric if both sides around a central value (like the mean) are mirror images. ⸻ 17. Q: Which common distributions are symmetric? A: Normal, Student’s t, Cauchy, uniform, and certain beta distributions. ⸻ 18. Q: When is a binomial distribution symmetric? A: Only when the probability of success p = 0.5 (like a fair coin toss). ⸻ 19. Q: What is a positively skewed distribution? A: Most values are concentrated on the left, with a long tail stretching to the right. ⸻ 20. Q: What is a negatively skewed distribution? A: Most values are concentrated on the right, with a long tail stretching to the left. ⸻ Would you like a second flashcard set just for Poisson, binomial, normal, uniform, and t-distributions (with formulas and conditions for use)?

Answer 14

Thanks! Based on this passage, here are flashcard questions only to help reinforce the concepts of skewness and distribution behavior: ⸻ 📉 SKEWNESS & DISTRIBUTIONS — FLASHCARDS (Questions Only) 1. Which two distributions are common examples of positively skewed distributions? 2. How does the value of p affect the skewness of a binomial distribution? 3. When is a binomial distribution positively skewed? 4. When is a binomial distribution symmetric? 5. When is a binomial distribution negatively skewed? 6. Can beta distributions be symmetric, positively skewed, or negatively skewed? What does this depend on? 7. What determines the shape of a beta distribution? 8. Why is it important to know if a distribution is symmetric or skewed? ⸻ Would you like the answers to these now, or shall I continue with another flashcard set (e.g., on distribution formulas or key properties)?

Answer 15

Estimation and Hypothesis Testing a. Estimation Definition: Using sample data to estimate a population value (like a mean or proportion). Types: • Point estimate (single value) • Interval estimate (range, like a confidence interval) Assumptions: Random sampling, reliable measurement. Limitations: Point estimates can be misleading without confidence intervals. Advantages: Gives an idea of population characteristics. Public Health Example: From a sample, you estimate 30–40% of children are underweight (95% confidence). Confidence interval is a interval estimation and used in hypothesis testing b. Hypothesis Testing Definition: A statistical method to test an assumption about a population. Terms: • Null hypothesis (H₀): no effect • Alternative hypothesis (H₁): there is an effect When used: To test new treatments, policies, trends. Assumptions: Depends on test—normality, independence, equal variances, etc. When not to use: With biased data, small samples, or when assumptions are violated. Limitations: Misinterpretation of p-values; false positives/negatives. Advantages: Basis of scientific decision-making.

Answer 16

Great! Here’s a detailed yet easy-to-understand breakdown of the topics you listed, including definitions, when to apply them, assumptions, limitations, advantages, and real-world public health examples: ⸻ 1. Parametric and Non-parametric Methods Category Parametric Methods Non-parametric Methods Definition Statistical tests that assume data comes from a population with a specific distribution (usually normal). Tests that don’t assume a specific distribution. When to Use - Large sample size- Data is normally distributed- Interval or ratio scale - Small samples- Data is not normally distributed- Ordinal or ranked data Examples t-test, ANOVA, Pearson correlation, Linear regression Mann-Whitney U, Wilcoxon, Kruskal-Wallis, Spearman correlation Assumptions - Normality- Equal variances- Independent observations - Fewer assumptions- Often based on medians/ranks Limitations Less flexible with skewed or small data Less powerful (needs larger effect to detect significance) Advantages More powerful if assumptions are met Flexible, works with ordinal or non-normal data Public Health Example Compare mean blood pressure between male and female adults using a t-test. Compare pain scores (ordinal) between two treatment groups using Mann-Whitney U test. ⸻ 2. Methods Used in Inferential Statistics and When to Apply Them Method Purpose When to Use Public Health Example Confidence Intervals Estimate a range for a population value Estimating means, proportions 95% CI for HIV prevalence in women Hypothesis Testing (t-test, ANOVA) Test a claim about a population Comparing means Does a new diet reduce cholesterol? Chi-square Test Association between categorical variables Comparing observed vs expected frequencies Handwashing vs diarrhea presence Regression (linear/logistic) Predict outcomes, adjust for factors Predicting or modeling outcomes Predict diabetes based on BMI, age Non-parametric tests When assumptions for parametric are not met Skewed or ordinal data Pain scores between two groups Correlation (Pearson/Spearman) Assess strength/direction of relationships Continuous (Pearson), ordinal (Spearman) Sugar intake vs BP levels ⸻ 3. Methods Used in Descriptive Statistics and When to Apply Them Method What it Describes When to Use Public Health Example Measures of Central Tendency Mean, median, mode Summarizing the center of data Mean age of tuberculosis patients Measures of Spread Range, variance, SD, IQR Show how data is spread SD of hemoglobin levels Frequency Tables Number of occurrences in categories For categorical data Frequency of malaria by region Graphs & Charts Pie, bar, histogram, boxplot Visualize data Pie chart of vaccination coverage Percentiles & Quartiles Distribution position Assessing health status Children in lowest height percentile ⸻ 4. Elementary Probability and Probability Distributions (Normal Distribution) Concept Definition When to Use Assumptions Example Probability Likelihood of an event (0–1) Predict risk, plan interventions Defined outcomes Chance a smoker develops lung cancer Probability Distribution Describes how values of a variable are distributed Modeling outcomes Depends on type: normal, binomial, etc. Height or weight distribution Normal Distribution Symmetric, bell-shaped curve For many natural, biological variables Continuous data, symmetrical Birth weights in a large sample Binomial Distribution For yes/no outcomes over trials Modeling events like infection (yes/no) Fixed # of trials, binary outcomes Probability of 3+ kids being anemic out of 5 ⸻ 5. Estimation and Testing of Hypotheses Component Definition When to Use Assumptions Example Point Estimate Single value estimate of population parameter Estimating mean/proportion Random sample 38% of women are anemic Confidence Interval Range that likely contains population parameter To reflect uncertainty in estimate Random, representative sample 95% CI for average hemoglobin level Null Hypothesis (H₀) Assumes no difference/effect Used in all hypothesis testing — “There’s no difference in drug A vs B” Alternative Hypothesis (H₁) What we suspect is true — — “Drug A is more effective” p-value Probability of observing result if H₀ is true Determines significance Valid test assumptions p < 0.05 suggests rejecting H₀ Type I Error (α) False positive p too small when H₀ true — Conclude vaccine works when it doesn’t Type II Error (β) False negative p too large when H₁ true — Miss real vaccine effect ⸻ 6. Chi-square, Regression, and Non-parametric Methods Method Definition When to Use Assumptions Limitations Advantages Public Health Example Chi-square Test Compares observed vs expected frequencies Categorical data (e.g. yes/no) Independent observations, expected ≥5 Doesn’t show strength/direction Easy to use for categorical data Link between education and contraceptive use Regression Models relationship between variables Predicting outcomes, adjusting for confounders Linearity, independence, normality (for linear) Outliers affect results Adjusts for many variables Predict HIV risk based on age, behavior Non-parametric Tests Compare medians/ranks without assuming normality Skewed, ordinal, or small data sets Fewer assumptions Less statistical power Robust and flexible Compare patient satisfaction scores (ordinal) across clinics ⸻ ✅ Summary Table Topic When to Use Assumptions Advantages Limitations Example Parametric Normal, continuous data Normality, equal variances Powerful, accurate Not suitable for skewed data Compare average BMI Non-parametric Skewed/ordinal/small data Minimal Flexible Less powerful Pain scores Inferential Stats To generalize from samples Random sample Informs decisions Can mislead if sample is biased Estimate anemia rate Descriptive Stats To summarize data None Quick overview No predictions Median age Probability To model risk Defined outcomes Helps in risk estimation Misunderstood easily Chance of infection Estimation & Hypothesis For inference/testing Sample reliability Generalizes findings Risk of false results Vaccine effect test Chi-square Association between categories Expected counts ≥5 Easy to use No strength measure Sanitation vs diarrhea Regression Prediction, control for variables Depends on type Predictive power Affected by outliers Predict diabetes risk Non-parametric Skewed or ranked data Few assumptions Works with most data Less sensitive Satisfaction rankings ⸻ Would you like this in a visual summary PDF or editable study sheet (like Google Docs or Word)?

Lecture 1 Ucc Flashcards

(44 cards)