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Psyc 235: Introduction to Statistics http://www.psych.uiuc.edu/~jrfinley/p235/ DON’T FORGET TO SIGN IN FOR CREDIT! Announcements (1of2) • Early Informal Feedback https://webtools.uiuc.edu/formBuilder/Secure?id=974 8379 Open until Sat March 15th • Special Lecture Thurs March 13th: Conditional Probability (incl. Law of Total Prob., Bayes’ Theorem) Mandatory for invited students Anyone can come No OH; Go to lab for Qs/help. Announcements (2of2) • Target Dates: STAY ON TARGET! You should be finishing the Distributions slice VoD “5. Normal Calculations, 17. Binomial Distributions,” and “18. The Sample Mean and Control Charts,” • Quiz 3: Thurs-Fri March 13th-14th Population “Standard Error” Sampling Distribution X (of the mean) n X Sample size = n X sample statistic (a random variable!) Shape of the Sampling Distribution? • If population distribution is normal: Sampling distribution is normal (for any n) • If sample size (n) is large: Sampling distribution approaches normal Central Limit Theorem • As sample size (n) increases: Sampling distribution becomes more normal Variance (and thus std. dev.) decreases Great, Normal Distributions! • Can now calculate probabilities like: • Just convert values of interst to z scores x (standard normal distribution) z • And then look up probabilities for that z score in ALEKS (calculator) • Or vice versa… So far… • We’ve been doing things like: Given a certain population, what’s prob of getting a sample statistic above/below a certain value? Population--->Sample • How can we shift to … Using our Sample to reason about the POPULATION? Sample--->Population INFERENTIAL STATISTICS! • Estimating a population parameter (e.g., the mean of the pop.: ) • How to do it: Take a random sample from the pop. Calculate sample statistic (e.g., the mean of the sample: ) X That’s your estimate. • Class dismissed. No, wait! • The sample statistic is a point estimate of X the population parameter by a little, or by a lot! • It could be off, Population Sampling Distribution (of the mean) X Sample size = n X We only have one sample statistic. And we don’t know where in here it falls. Interval Estimate • Point estimate (sample statistic) gives us no idea of how close we might be to the true population parameter. • We want to be able to specify some interval around our point estimate that will have a high prob. of containing the true pop parameter. Confidence Interval • An interval around the sample statistic that would capture the true population parameter a certain percent of the time (e.g., 95%) in the long run. (i.e., over all samples of the same size, from the same population) Note: True Population Parameter is constant! This is the mean from one sample. Let’s put a 90% Confidence Interval around it. X Note that this particular interval captures the true mean! Let’s consider other possible samples (of the SAME SIZE) X So does this one. This one too. This interval misses the true mean! The mean from another possible sample. This one captures the true mean too. And this one. Yep. … But this one’s alright. A 90% Confidence Interval means that for 90% of all possible samples (of the same size), that interval around the sample statistic will capture the true population parameter (e.g., mean). X Only sample statistics in the outer 10% of the sampling distribution have confidence intervals that “miss” the true population parameter. … But, remember… X … But, remember… All that we have is our sample. Sample size = n X Still, a Confidence Interval is more useful in estimating the population parameter than is a mere point estimate alone. So, how do we make ‘em? Sample size = n X CONFIDENCE INTERVAL (1 - )% confidence interval for a population parameter P( C. I. encloses true population parameter ) = 1 - Note: = P(Confidence Interval misses true population parameter ) “Proportion of times such a CI misses the population parameter” Margin of Error Point estimate ± sample statistic ex: X z / 2 critical value or · t / 2 Std. dev. of point estimate standard deviation of sampling distribution (aka “Standard Error”) Decision Tree for Confidence Intervals Population Standard Deviation known? Yes Pop. Distribution normal? n large? (CLT) Yes No Standard normal distribution Yes No No Note: ALEKS… Critical Score z-score Yes z-score Can’t do it t-score t distribution No Yes No t-score Can’t do it C.I. using Standard Normal Distribution For the Population Mean When known. First, choose an level. For ex., α=.05 gives us a 95% confidence interval. Margin of Error Point estimate ± critical value · Std. dev. of point estimate C.I. using Standard Normal Distribution For the Population Mean When known. First, choose an level. For ex., α=.05 gives us a 95% confidence interval. Margin of Error X ± critical value · Std. dev. of point estimate C.I. using Standard Normal Distribution For the Population Mean When known. First, choose an level. For ex., α=.05 gives us a 95% confidence interval. Margin of Error X ± critical value · n C.I. using Standard Normal Distribution For the Population Mean When known. First, choose an level. For ex., α=.05 gives us a 95% confidence interval. Margin of Error X ± z / 2 · n Lookup value (ALEKS calculator, Z tables) Handy Zs (Thanks, Standard Normal Distribution!) if .10 90% Confidence upper .05 z / 2 1.645 critical value if .05 95% Confidence upper .025 z / 2 1.960 critical value if .01 99% Confidence upper .005 z / 2 2.576 critical value C.I. using Standard Normal Distribution For the Population Mean When known. Margin of Error X ± z / 2 · n X z / 2 is a 1 confidence interval of n Remember: random variable Furthermore, in that case, PX z / 2 X z / 2 1 n n C.I. using t Distribution For the Population Mean When unknown! Margin of Error X ± · C.I. using t Distribution For the Population Mean When unknown! Margin of Error X · ± s n We use the standard deviation from our sample (s) to estimate the population std. dev. (). s x x 2 i n 1 The “n-1” is an adjustment to make s an unbiased estimator of the population std. dev. C.I. using t Distribution For the Population Mean When unknown! Margin of Error X ± t / 2 · s n Critical value taken from a t distribution, not standard normal. The goodness of our estimate of will depend on our sample size (n). So the exact shape of any given t distribution depends on degrees of freedom (which is derived from sample size: n-1, here). Fortunately, we can still just LOOK UP the critical values… (just need to additionally plug in degrees freedom) Behavior of C.I. • As Confidence (1-) goes UP Intervals get WIDER (ex: 90% vs 99%) • As Population Std. Dev. () goes UP Intervals get WIDER • As Sample Size (n) goes UP Intervals get NARROWER n Std dev of sampling distribution of the mean C. I. for Differences (e.g., of Population Means) • Same approach. • Key is: Treat the DIFFERENCE between sample means as a single random variable, with its own sampling distribution & everything. X1 X 2 The difference between population means is a constant (unknown to us). Remember • Early Informal Feedback • Special Lecture Thursday No OH; Go to lab for Qs/help. • Stay on target Finish Distributions VoDs • Quiz 3 X