Review of Chapters 1-5.ppt
- 1. Review of Chapters 1- 5 We review some important themes from the first 5 chapters 1. Introduction • Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods for • Design – Planning / Implementing a study • Description – Graphical and numerical methods for summarizing the data • Inference – Methods for making predictions about a population (total set of subjects of interest, real or conceptual), based on a sample
- 2. 2. Sampling and Measurement • Variable – a characteristic that can vary in value among subjects in a sample or a population. Types of variables • Categorical • Quantitative • Categorical variables can be ordinal (ordered categories) or nominal (unordered categories) • Quantitative variables can be continuous or discrete • Classifications affect the analysis; e.g., for categorical variables we make inferences about proportions and for quantitative variables we make inferences about means
- 3. Randomization – obtaining reliable data by reducing potential bias Simple random sample: In a sample survey, each possible sample of size n has the same chance of being selected. Randomization in a survey used to get a good cross- section of population. With such probability sampling methods, standard errors tell us how close sample statistics tend to be to population parameters. (Otherwise, the sampling error is unpredictable.)
- 4. Experimental vs. observational studies • Sample surveys are examples of observational studies (merely observe subjects without any experimental manipulation) • Experimental studies: Researcher assigns subjects to experimental conditions. – Subjects should be assigned at random to the conditions (“treatments”) – Randomization “balances” treatment groups with respect to lurking variables that could affect response (e.g., demographic characteristics, SES), makes it easier to assess cause and effect
- 5. 3. Descriptive Statistics • Numerical descriptions of center (mean and median), variability (standard deviation – typical distance from mean), position (quartiles, percentiles) • Bivariate description uses regression/correlation (quantitative variable), contingency table analysis (categorical variables). Graphics include histogram, box plot, scatterplot
- 6. •Mean drawn toward longer tail for skewed distributions, relative to median. •Properties of the standard deviation s: • s increases with the amount of variation around the mean •s depends on units of the data (e.g. measure euro vs $) •Like mean, affected by outliers •Empirical rule: If distribution approx. bell-shaped, about 68% of data within 1 std. dev. of mean about 95% of data within 2 std. dev. of mean all or nearly all data within 3 std. dev. of mean
- 7. Sample statistics / Population parameters • We distinguish between summaries of samples (statistics) and summaries of populations (parameters). Denote statistics by Roman letters, parameters by Greek letters: • Population mean =m, standard deviation = s, proportion are parameters. In practice, parameter values are unknown, we make inferences about their values using sample statistics.
- 8. 4. Probability Distributions Probability: With random sampling or a randomized experiment, the probability an observation takes a particular value is the proportion of times that outcome would occur in a long sequence of observations. A probability distribution lists all the possible values and their probabilities (which add to 1.0)
- 9. Like frequency dist’s, probability distributions have mean and standard deviation Standard Deviation - Measures the “typical” distance of an outcome from the mean, denoted by σ ( ) ( ) E Y yP y m
- 10. Normal distribution • Symmetric, bell-shaped • Characterized by mean (m) and standard deviation (s), representing center and spread • Prob. within any particular number z of standard deviations of m is same for all normal distributions • An individual observation from an approximately normal distribution satisfies: – Probability 0.68 within 1 standard deviation of mean – 0.95 within 2 standard deviations – 0.997 (virtually all) within 3 standard deviations
- 11. Notes about z-scores • z-score represents number of standard deviations that a value falls from mean of dist. • A value y is z = (y - µ)/σ standard deviations from µ • The standard normal distribution is the normal dist with µ = 0, σ = 1
- 12. Sampling dist. of sample mean • is a variable, its value varying from sample to sample about population mean µ. Sampling distribution of a statistic is the probability distribution for the possible values of the statistic • Standard deviation of sampling dist of is called the standard error of • For random sampling, the sampling dist. of has mean µ and standard error popul. std. dev. sample size y n s s y y y y y
- 13. Central Limit Theorem: For random sampling with “large” n, sampling dist of sample mean is approximately a normal distribution • Approx. normality applies no matter what the shape of the population distribution • How “large” n needs to be depends on skew of population dist, but usually n ≥ 30 sufficient • Can be verified empirically, by simulating with “sampling distribution” applet at www.prenhall.com/agresti. Following figure shows how sampling distribution depends on n and shape of population distribution. y
- 15. 5. Statistical Inference: Estimation Point estimate: A single statistic value that is the “best guess” for the parameter value (such as sample mean as point estimate of popul. mean) Interval estimate: An interval of numbers around the point estimate, that has a fixed “confidence level” of containing the parameter value. Called a confidence interval. (Based on sampling dist. of the point estimate, has form point estimate plus and minus a margin of error that is a z or t score times the standard error)
- 16. Confidence Interval for a Proportion (in a particular category) • Sample proportion is a mean when we let y=1 for observation in category of interest, y=0 otherwise • Population prop. is mean µ of prob. dist having • The standard deviation of this prob. dist. is • The standard error of the sample proportion is (1) and (0) 1 P P (1 ) (e.g., 0.50 when 0.50) s ˆ / (1 )/ n n s s ˆ
- 17. Finding a CI in practice • Complication: The true standard error itself depends on the unknown parameter! ˆ / (1 )/ n n s s In practice, we estimate and then find 95% CI using formula ^ ^ ^ 1 (1 ) by se n n s ˆ ˆ 1.96( ) to 1.96( ) se se
- 18. CI for a population mean • For a random sample from a normal population distribution, a 95% CI for µ is where df = n-1 for the t-score • Normal population assumption ensures sampling dist. has bell shape for any n (Recall figure on p. 93 of text and next page). Method is robust to violation of normal assumption, more so for large n because of CLT. .025( ), with / y t se se s n
- 19. Questions you should be able to answer: • What is the difference between descriptive statistics and inferential statistics? Give an example of each. • Why do we distinguish between quantitative variables and categorical (nominal and ordinal) variables? • What is a simple random sample? • Why should a survey use random sampling, and why should an experiment use randomization in assigning subjects to treatments? • What is the difference between an experiment and an observational study? • Define the mean and median, and give an advantage and disadvantage of each. Give an example when they would be quite different? Which would be larger, and why? • How do you interpret a standard deviation?
- 20. • Which measures are resistant to outliers: Mean, median, range, standard deviation, inter-quartile range? • What information does a box plot give you? • How can you describe an association between two (i) quantitative variables, (ii) categorical variables • What is the correlation used for, and how do you interpret its value? • If you know or can estimate P(A) and P(B), how do you find P(A and B) when A and B are independent? • How do you use the normal distribution to get tail probabilities, central probabilities, z-scores? • What does the z-score represent? • What is a sampling distribution? • What is a standard error? • What’s the difference between s, σ, se. Which is the smallest? • Why is the normal distribution important in statistics?
- 21. • What does the Central Limit Theorem say? Why is it important in statistical inference? • What are the differences among (i) population distribution, (ii) sample data distribution, (iii) sampling distribution? • How do you interpret a confidence interval? • Why do standard errors have to be estimated, and what impact does that have (i) for inference about a proportion, (ii) for inference about a mean? • How does the t distribution depend on the df value, and how does it compare to the standard normal distribution? • How does a confidence interval depend on the (i) sample size, (ii) confidence level? How do you find the z-score or t- score for a particular confidence? • Why do you use a t-score instead of a z-score when you find the margin of error for a confidence interval for a mean?