Stat 217 – Day 13 Sampling Distributions (Topic 13) Submit Activity 12-6?

Last Time –Normal Distribution(p.245) The normal distribution provides a reasonable probability model for many quantitative variables  Biological characteristics (temperatures, body weights, footprint lengths, blood pressure, pulse rates, measurement errors) Calculate probabilities above, below, between  Technology (Normal Probability Calculator) OR Standardize (z-score) and use Table II Calculate a percentile given a probability  E.g., lightest 2.5% of birthweights

Activity 12-2 Rule #1: Start with a sketch  Labeling essential, reasonable scaling helpful Activity 13-1

Last Time – Activity 12-2 (f) Pr(weight > 4536g) With Table II, take value given (probability below) and subtract from one OR, look up z = About 1.5% of babies weigh more than 10 lbs

Last Time – Activity 12-2 (h) Between 3000 and 4000 (middle) Expect about 59% of babies to be in this range (i) Actual data: 2,697819/ =.656 Low birthweight: / =.081 With Table II, find each z- value, convert each to the probability below, and subtract -

Last Time – Activity 12-2 Bottom 2.5%  Based on empirical rule: 2 standard deviations  Based on normal distribution: Start with probability and solve for “observation” With Table II, look in middle of table for closest probability value, read off the corresponding z-value (including hundredths place) and convert to “x” value 3300 – 1.96(2500) = grams

Last Time – Activity 12-2 (k)Top 10% aka Bottom 90% A baby would need to weight or more grams to be in the top 10% ( to weigh more than 90% of newborns)

Kissing Couples How often do you think kissing couples turn to the right? Suppose you thought it was 50/50 but you found a sample result of 64.5%, would this convince you that kissing couples in the entire population tend to turn right more than left?

Topic 13 (p. 252) Recall: Parameters are numbers summarizing a characteristic of the population; statistics are numbers summarizing a characteristic of the sample  Means, proportions Here come the symbols!

Activity 13-1 Take a sample of 25 candies  Two people per bag Determine the count (how many) of each color and the proportion of orange Answer questions (b)-(h) I predict all but 1 or 2 of you will be between.3 and.7.

What do we learn? Sample proportions vary from sample to sample But they have a predictable pattern Do you think the normal distribution can model the behavior of sample proportions?  What is the mean?  What is the standard deviation?

Activity 13-2: Long-run pattern Reeses Pieces applet  Course Materials > Stat 217 Java applets  Answer (a)-(g) and then (k)-(l)

Activity 13-2 Sampling Distribution 95% of sample proportions fall within.20 of the population proportion 95% of students will get a sample proportion within.20 of the population proportion  Regardless of the value of 

Activity 13-2: Sample Size? (k) Mean =.45, SD .06 (l) Shape and center the same, spread much less (more precision) (o) In general, sample proportion is more likely to be closer to the population proportion with the larger sample size Keep in mind, roughly 95% of sample proportions should be within 2  SD of population proportion where SD depends on sample size

The Central Limit Theorem(p. 259) When you take a random sample from a population with proportion , If the sample size is large, then the sampling distribution of the sample proportions will follow a normal distribution with mean equal to the population proportion  and standard deviation equal to SD( ) =

Verification (s) n = 25, pi =.45 Sample size large enough? Mean and SD? (t) n = 75, pi =.45 Sample size large enough? Mean and SD? (u) n = 25, pi =.10 Sample size large enough? Mean and SD?

To turn in individually – IN BLACKBOARD 1. What was your sample proportion orange? 2. If given the choice which would you prefer to hear first, good news or bad news? 3. If asked to predict the outcome of a coin toss, do you predict heads or tails? 4. Which cat was mine For Wednesday  Self-check Activity 13-4  Lots of Watch Out points p  Be finishing Lab 4  HW 4 now posted