Expected values, standard errors, Central Limit Theorem. Statistical inference

Transcription

1 Expected values, standard errors, Central Limit Theorem FPP Statistical inference Up to this point we have focused primarily on exploratory statistical analysis We know dive into the realm of statistical inference The ideas associated with sampling distributions, p-values, and confidence intervals are more abstract and are therefore slightly harder These concepts are also very powerful For good if used correctly For bad if used incorrectly 1

2 Statistics vs probability modeling Probability: know the truth, want to estimate the chances that data occur Statistics: know the data that occur, want to infer about the truth Parameter Population Inference Statistic x Sample Law of averages What does the law of averages say? Toss a coin As # of tosses increase the #heads 0.5(#tosses %heads 50% 2

3 Chance processes When tossing a coin: Actual #heads Expected #heads What is the likely size of the difference? Strategy: Find an analogy between the process being studied and drawing numbers at random from a box (box model) Box models A so called box model is a good starting point into statistical inference The purpose of these very simple models is to analyze chance variability They are a construction for learning about characteristics of populations 3

4 Motivating example Population: 119,106 graduates of Duke Variable: donation amount in $$ to Duke Annual Fund in 2001 Box model: make a ticket for every alumnus containing his/her donation amount Put all these tickets in a hypothetical box. Box models: typical questions Pick 100 tickets at random from the box, with replacement 1. Before collecting the data, what do you expect the sum of these 100 alumni donations to equal? 2. What do you think is a typical deviation from this expected value? 3. Before collecting the data how many of the 100 alumni people do you expect to be donators? 4. What do you think is a typical deviation from this expected value? 4

5 Characteristics of alumni donations For the 119,106 alumni: Average of all donations = $735 SD of donations = $23,827 42,938 donated (36%) 76,168 did not donate (64%) Learning about the sample sum When we sample randomly, the sum of the 100 tickets will differ for different samples What is the expected value (EV) of the sample sum E(sample sum) = n*(average of box) What is a typical deviation of a sample sum from this expected value Standard error (SE) of sum = n *(SD of box) 5

6 Sample sum of donations for 100 alumni So the sum of the 100 alumni donations should be: E(sample sum) = 100*($735) = $73,500 give or take the SE SE = 100($23,827) = $238,270 How sure are we about the sum of donations using a sample of 100? Key idea If we take independent samples of 100 alumni over and over again, recording each sample then The average of the sample sums should be around $73,500 The SD of the sample sums should be around $238,270 Box model for binary (dichotomous) outcomes 42,938 donated and 76,168 did not Make a box with tickets comprised of 42,938 ones and 76,168 zeros. Average of box = % of ones = 0.36 = p SD of box = 0.48 Short cut for SD for binary box models (and only for binary box models) Sample 100 tickets out of the box with replacement. What does this process remind you of? 6

7 Sample number of donators out of 100 alumni The number of donators in the sample equals the sample sum of the 0-1 tickets Thus, the expected number of donators is EV of sample sum = n * (Average of box) = 100 * 0.36 = 36 The typical deviation of the sample sum for expected value is The Standard error (SE) of sum = * (SD of box) n = 10 *.48 = 4.8 Sample number of donators out of 100 alumni Hence, the number of alumni who donated out of a random sample of 100 should be 36, give or take around 5 people (SD = 4.8). Compared to the average donation per alumni how confident are we that any give sample of 100 will produce 36 donors. Key idea If we take independent samples of 100 alumni over and over again, recording the number of donators in each sample The average of the sample number of donators should be around 36 The SD of the sample numbers of donators should be around 4.8 7

8 A problem from the text 100 draws are made with replacement from a box containing the seven numbers Suppose you were betting. The closer your guess is to the sample sum, the more money you win. What number would you guess? How much would you expect the sample sum to be off from the expected value of the sum? Difference between SD and SE SD is the typical deviation from the average in a box. SD is a property of the box; it doesn t depend on a random sampling SE is the typical deviation from the expected value in a random sample. SE results from random sampling SE gives an idea of how large the chance error is Sum of draws is likely to be around its expected value, but to be off by a chance error similar in size to its SE Sum of draws = EV ± chance error 8

9 EV and SE of the sample average or percent Since sample average(percent) = sample sum /n we get 1. Just like sample sums, sample averages and sample percentages are subject to chance variation 2. EV for sample average ( or %) = EV of sample sum / n = Avg. of box. 3. SE for sample average (or %) = SE for sample sum / n = SD of box / n Common theme for SE of sample average and sample percentage Fir a binary variable, the population SD = So both the sample average and sample percentage have a standard error of the form SE = Population SD / n 9

10 Sample averages and percentages In a random sample of 100 alumni, we expect the sample average donation In a random sample of 100 alumni, we expect the sample average donation to equal $735, give or take $2, We expect 36% to donate, give or take 4.8% (SE =.048). If we take independent samples of 100 alumni over and over again, recording the average donation and the percentage of donators in each sample: a) the average of the sample averages of donations should be around 735. b) the SD of the sample averages of donations should be around 2, c) the average of the sample percentages of donators should be around.36. d) the SD of the sample percentages of donators should be around Sample averages and percentages In a random sample of 100 alumni, we expect the sample average donation to equal $735 give or take $2, We expect 36% to donate, give or take 4.8% If we take independent samples of 100 alumni over and over again, recording the average donation and the percentage of donators in each sample The average of the sample averages of donations should be around 735 The SD of the sample averages of donations should be around 2, The average of the sample percentages of donators should be around 0.36 The SD of the sample percentages of donators should be around

11 Law of averages Plot the SE of sample average donation for an increasing sample taken from the box As n in increases, the SE of the sample average decreases This is called the law of averages Vegas was built on this law Shape of chance process The expected value and the standard error provide a measure of center and spread for the chance process What about the shape Book introduces something called the probability histogram This is a histogram of the samples take from the box model. What shape will this histogram take on 11

12 The central limit theorem Take many random samples from a box model, all of the samples of size n. When n is sufficiently large, the distribution of the sample average (or sample %) is welldescribed by a normal curve The mean of this normal curve is the EV and the standard deviation for this normal curve is the SE The Central Limit Theorem What does the CLT give us? A ton of stuff We can find probabilities and percentiles using the the normal table Can predict fairly accurately how unlikely it is to sample an observed sample mean Can assess rather accurately how likely a population mean lies within an interval 12

13 Central Limit Theorem What happens if the distribution of the original variable is not symmetric The central limit theorem still kicks in (the sample size n just needs to be bigger) What happens if the distribution of the original variable is bimodal The central limit theorem still kicks in (the sample size n just needs to be bigger) This is absolutely a fantastic result!!! Central Limit Theorem M&Ms Pick 50 M&Ms at random (from a bag). How likely is it to have less than 40% yellow and brown M&Ms in the bag? Assume 50% of all M&M s are yellow and brown (source: M&M s home page) For a sample proportion of yellow and brown M&Ms EV = 0.5 and SE = 13

14 Size of sample For binomial data, the CLT usually kicks in pretty well when both of the following conditions on sample size are met In practice we don t know p so we will plug in the sample percentage CLT and M&Ms Since n=50, CLT applies The probability of getting less than 40% yellow and brown M&Ms in a bag of 50 is It is somewhat unusual to get less than 40% yellow and brown M&Ms (about 8 chances in 100) 14

15 CLT household example The average size of U.S. households is 2.6 people. The SD of household size is (These are true values from the U.S. Census). Pick 200 houses at random in the U.S. How likely is it that we ll get a sample average household size of 3 or more? CLT household example For a sample average of 200 households EV = 2.6 and SE = The chance of getting an average household size greater than 3 equals the area under the standard normal curve to the right of 4. This is a very small chance 15

16 Alumni donations example In a random sample of 100 alumni, what is the chance that more than half donated? Alumni donations example What is the chance that the sample average of donations from 100 randomly picked alumni will be between $50 and $100 16

17 CLT under three conditions 1. If original variable follows a normal distribution the CLT holds regardless of sample size 2. If distribution of original variable is symmetric and unimodal then CLT holds for a small sample size (say less than 15) 3. If distribution is skewed, not unimodal then the CLT holds after a larger sample size how large depends on the sharpness of the skew. In this class we will follow convention and say