1STA 291 - Lecture 14 1STA 291Lecture 14, Chap 9• Sampling Distributions– Sampling Distribution of a sample statistic like the (sample) Proportion• Population has a distribution, which is fixed, usually unknown. (but we want to know)• Sample proportion changes from sample to sample (that’s sampling variation). Sampling distribution describes this variation.STA 291 - Lecture 14 2• Suppose we flip a coin 50 times and calculate the success rate or proportion. • same as asking John to shoot 50 free throws (biased coin).• Each time we will get a slightly different rate, due to random fluctuations. • More we flip (say 500 times), less the fluctuationSTA 291 - Lecture 14 32• How to describe this fluctuation?• First, use computer to simulate……• Repeatedly draw a sample of 25, etc• Applet:STA 291 - Lecture 14 4STA 291 - Lecture 14 5Sampling distribution of proportionSTA 291 - Lecture 14 6Sampling distribution: n=253STA 291 - Lecture 14 7Sampling distribution: n=100• Larger the n, less the fluctuation.• Shape is (more or less) symmetric, bell curve. STA 291 - Lecture 14 8Population distribution vs. sampling distribution• For example, when population distribution is discrete, the sampling distribution might be (more or less) continuous. • Number of kids per family is a discrete random variable (discrete population), but the sample mean can take values like 2.5 (sampling distribution is continuous).STA 291 - Lecture 14 94STA 291 - Lecture 14 10Sampling Distribution: Example Details• Flip a fair coin, with 0.5 probability of success (H). Flip the same coin 4 times.• We can take a simple random sample of size 4 from all sta291 students. • find if the student is AS/BE major.• Define a variable X where X=1 if the student is in AS/BE, (or success)and X=0 otherwise • Use the number “1” to denote success• Use the number “0” to denote failureSTA 291 - Lecture 14 11STA 291 - Lecture 14 12Sampling Distribution: Example (contd.)• If we take a sample of size n=4, the following 16 samples are possible:(1,1,1,1); (1,1,1,0); (1,1,0,1); (1,0,1,1);(0,1,1,1); (1,1,0,0); (1,0,1,0); (1,0,0,1);(0,1,1,0); (0,1,0,1); (0,0,1,1); (1,0,0,0);(0,1,0,0); (0,0,1,0); (0,0,0,1); (0,0,0,0)• Each of these 16 samples is equally likely (SRS !) because the probability of being in AS/BE is 50% in all 291students5STA 291 - Lecture 14 13Sampling Distribution: Example (contd.)• We want to find the sampling distribution of the statistic “sample proportion of students in AS/BE” • Note that the “sample proportion” is a special case of the “sample mean” • The possible sample proportions are 0/4=0, 1/4=0.25, 2/4=0.5, 3/4 =0.75, 4/4=1• How likely are these different proportions?• This is the sampling distribution of the statistic “sample proportion” STA 291 - Lecture 14 14Sampling Distribution: Example (contd.)Sample Proportion of Students from AS/BEProbability0.00 1/16=0.06250.25 4/16=0.250.50 6/16=0.3750.75 4/16=0.251.00 1/16=0.0625STA 291 - Lecture 14 15• This is the sampling distribution of a sample proportion with sample size n=4 and P(X=1)=0.56STA 291 - Lecture 14 163 key features• The shape getting closer to “bell-shaped”, almost continuous – become “Normal”• The center is always at 0.5 – (the population mean)• The variance or SD reduces as sample size n gets larger.Sample proportionSTA 291 - Lecture 14 17ˆpSTA 291 - Lecture 14 18Mean of sampling distribution• FACT: Mean/center of the sampling distribution for sample mean or sample proportion always equal to the same for all n, and is also equal to the population mean/proportion.ˆppµ=7STA 291 - Lecture 14 19Reduce Sampling Variability• The larger the sample size n, the smaller the variability of the sampling distribution• The SD of the sample mean or sample proportion is called Standard Error• Standard Error = SD of population/ nSTA 291 - Lecture 14 20Normal shape• Shape becomes normal type. -- Central Limit Theorem STA 291 - Lecture 14 21Interpretation• If you take samples of size n=4, it may happen that nobody in the sample is in AS/BE• If you take larger samples (n=25), it is highly unlikely that nobody in the sample is in AS/BE• The sampling distribution is more concentrated around its mean• The mean of the sampling distribution is the population mean: In this case, it is 0.58STA 291 - Lecture 14 22• The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviationSTA 291 - Lecture 14 23Standard Error• Intuitively, larger samples yield more precise estimates• Example: – X=1 if student is in AS/BE, X=0 otherwise– The population distribution of X has mean p=0.5 and standard deviation (1)0.5pp−=STA 291 - Lecture 14 24Standard Error• Example (contd.): – For a sample of size n=4, the standard error of is – For a sample of size n=25,– Because of the approximately normal shape of the distribution, we would expect to be within 3 standard errors of the mean (with 99.7% probability)X0.50.254Xnσσ ===0.50.125Xnσσ ===X9STA 291 - Lecture 14 25Central Limit Theorem• For random sampling (SRS), as the sample size n grows, the sampling distribution of the sample mean approaches a normal distribution• Amazing: This is the case even if the population distribution is discrete or highly skewed• The Central Limit Theorem can be proved mathematically• We will verify it experimentally in the lab sessionsXSTA 291 - Lecture 14 26Attendance Survey Question• On a 4”x6” index card–Please write down your name and section number–Today’s Question: Do you believe in the “Hot hand” claim?STA 291 - Lecture 14 27Extra: March Madness• Some statistics related to sports: how does a ranking system for basketball team work? Also the ranking of tennis players, chess players, etc.10STA 291 - Lecture 14 28Theory of “hot hand”• One theory says, if a player gets hot (hit several 3 point in a row) then he is more likely to hit the next one. Because he has a hot hand……STA 291 - Lecture 14 29Effect of Sample Size• The larger the sample size n, the smaller the standard deviation of the sampling distribution for the sample mean– Larger sample size = better precision • As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution– Usually, for about n=30 or above, the