Sociology 30 and Central Limit Theory Liying Luo 04.05 Schlitz vs. Michelob Super Bowl XV Commercial Why was Schlitz so sure that the results aired live at Super Bowl halftime would not screw them? Inferential Statistics We actually observe a Sample of cases a Population of Cases We want to draw conclusions about a Sample of cases Descriptive Statistics (Why would we do this?) a Population of Cases Inferential Statistics Descriptive Statistics
Inferential Statistics Popula(on Parameter: numeric afributes of a populagon Sample Sta(s(cs: numeric afributes of a sample a populagon parameter or sample stagsgcs: Mode of the SEXFREQ in the GSS? Standard deviagon of the height of the students in the room? Average age in the 200 Census data? Inferential Statistics In the Schlitz vs Michelob example Population? Population parameter? Sample? Sample statistics? Inferential Statistics A CNN/ORC poll conducted a poll among 248 registered voters who described themselves as republicans on March 7-20. Candidate Percentage Trump 47% Cruz 3% KaGch 7% Someone else 3% None/no one % No opinion % In this example Population? Population parameter? Sample? Sample statistics?
Recall the definigon distribugon... Consider taking one random sample consisgng of 5 students from a class and calculagng the proporgon of preferring Pepsi in the sample. Then take another random sample consisgng of 5 students from the same class and calculate the proporgon of preferring Pepsi. Would you expect both proporgons to be exactly the same? The distribugon of a sample stagsgc has a special name Sampling Distribu(on: the distribugon of sample stagsgcs (CLT): If n (sample size) is sufficiently large, then the sample means from many random samples from a populagon with mean μ and variance σ 2 are approximately normally distributed with mean μ and variance σ 2 /n regardless of the distribugon of the original variable. Sampling distribu(on vs sample distribu(on
Sample Distribution Draw a random sample of 50 college students and record Sociology majors. What does the distribugon of these 50 observagons look like? Draw a random sample of 0 college students and record their GPA. What does the distribugon of these 0 GPAs look like? Sample Distribution Sample distribugon: distribugon of observagons in one sample Random Sample Random Sample of n=0 college students 50 2 40 30 20 0 0 No Sociology Major? es 0 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 Sample Distribution Sample distribugon: distribugon of observagons in one sample Random Sample Random Sample of n=0 college students 50 2 40 30 20 0 0 No Sociology Major? es 0 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 Sample Percentage = 4.0% Sample Mean = 3.0
Draw a random sample of 50 college students and calculate the proporgon of Sociology majors. Repeat the process,000 Gmes. What does the distribugon of the,000 proporgons look like? Draw a random sample of 0 college students and calculate their average GPA. Repeat the process 0,000 Gmes. What does the distribugon of the 0,000 average GPAs look like? Random Sample Random Sample of n=0 college students Sample Percentage = 4.0% Sample Mean = 3.0 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 5 5 of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7
20 20 of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 00 00 of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7,000,000 of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7
00,000 00,000 of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 Infinity Infinity of n=0 college students 0% 4% 8% 2% 6% 20% 24% 28% 32% 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 (CLT): If n is sufficiently large, then the sample means from many random samples from a populagon with mean μ and variance σ 2 are approximately normally distributed with mean μ and variance σ 2 /n regardless of the distribugon of the original variable.
From the 880 Census, the population distributions of Born in the United States? Number of People in the Household True Population Mean µ = 5.3 True Population Proportion p = 0.8668 Number of People in the Household Born in the US? (0=No, =es). I sampled 00 individuals (n=00), and computed the sample proporgon born in the U.S. the sample mean number of people in the household 2. I repeated that procedure 999 more Gmes. sample distribution of # of people sampling distribution of average # of people # of people in the household
Binomial Random Variables Binomial Random Variables For binomial random variable based on n trials with probability of success on any given trial equal to p, the For binomial expected random value of variable (or µ ) equals based on n trials with probability E() of = success μ = np on any given trial equal to p, the expected value sample of (or distribution µ of sampling distribution of the variance σ 2 birth equals ) equals place proportion of native-born E() = μ σ 2 =np( = np -p) the variance σ 2 equals Born in the US? and the standard deviation σ equals σ 2 =np( -p) 2 σ = σ = np( -p) and the standard deviation σ equals 2 σ = σ = np( -p) Binomial Random Variables Binomial Random Variables Because a binomial random variable is just a special case of discrete random variables, the formula used to compute Because Central the mean a Limit binomial Theorem and variance random (CLT): of variable Using a discrete sample is just random a stagsgcs special variable to case esgmate of populagon parameters. discrete random variables, k the formula used to compute the mean E() and = μ variance = p( ) i i i= of a discrete random variable : sample k mean 2 2 k σ = ( μ p() i i se E() : standard = μ = i p error; ( ) i i standard deviagon of the sample mean. i= k 2 will 2 also work for binomial random variables (if you are inclined to σ = ( μ ) p() i i do i= extra math) will also work for binomial random variables (if you are inclined to do extra math) Binomial Random Variables Binomial Random Variables Whether a newborn baby is a boy is a binomial experiment with p = 0.52 Whether Central For the a newborn Limit binomial baby Theorem random is a boy variable is a binomial number experiment of boys based with on p = n 0.52 = 00 randomly selected births: For the E() binomial = μ = np 00x 0.52 5.2 random = variable = number of boys based on When n = 00 σthe 2 = randomly populagon np(-p) = selected 00(0.52)( standard births: deviagon 0.52) is = known 24.99 E() = s se σμ 2 = = np σ = 00x np(-p) 0.52= = 5.2 24.99 = 4.999 σ 2 = np(-p) n = 00(0.52)( 0.52) = 24.99 If we sampled 00 births and counted the number of boys 2 σ : = standard σ then = np(-p) deviagon repeated = of that 24.99 the experiment populagon = 4.999 an infinite number of If we n times we sampled : size 00 would births average and counted 5.2 boys the with number σ = 4.999 of boys and then repeated that experiment an infinite number of times we would average 5.2 boys with σ = 4.999 Sociology 38 ~ February 0, 205 3 Sociology 38 ~ February 0, 205 3 Sample Estimate ± ± t a/2 s n
Recap Probability: How likely a pargcular outcome of an event will occur Normal Distribu(on: Z-Score: the number of standard deviagons an observagon is above or below the mean. i Zi s Recap μ 6 6 ou select one sample of n = 0 college students. What is the probability that your sample mean GPA differs from the populagon mean by more than ±0.4? μ Infinity of n=0 college students 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 for Mean GPA µ = 3.0 σ = 0.2
Born in the United States? Worksheet 95% of the chance that the sample mean will fall within.96 standard deviagons of the populagon mean. ou select one sample of n=0 college students 6 6 Infinity of n=0 college students μ What is the probability that your sample mean GPA ( x ) differs from the population mean by more than ±0.4? 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 for Mean GPA µ = 3.0 σ = 0.2 Worksheet ou select one sample of n=0distribution college students Sampling and Infinity your sample mean GPA ( x ) is greater than 3.5? of n=0 college students is the probability SchlitzWhat vs. Michelob Super Bowlthat XV Commercial 2. 2.3 2.5 2.7 2.9 3. 3.3 3.5 3.7 for Mean GPA µ = 3.0 σ = 0.2 Soc 38-2/7/205 Worksheet 00 Michelob drinkers were randomly selected to conduct a blind taste test between Michelob and Schlitz. Why was Schlitz so sure that the results aired live at Super Bowl haloime would not screw them? 6
Exercise ou selected one random sample of,000 people and got a sample mean of 5 for the outcome of interest. It is known that standard deviagon of the populagon 20. What is the probability that your sample mean differs from populagon mean by more than 5?