Sampling Distributions Chapter 7 Sampling Distributions

Sampling Distributions Sampling distribution of a statistic - the probability distribution of the statistic computed from all possible random samples using the same sample size from the sample population

Sampling Distributions (broken down) Sample - subset of scores from all the scores (number of observations = n) Random sample - each observation has an equal chance of being included Statistic - quantitative characteristic of a sample ( M, s2, s, etc.) Probability Distribution - represents scores and the likelihood of their occurrence (graphically, in tables)

Sampling Distributions Shows the likelihood of obtaining the value of some statistic given n observations What is the probability of getting a mean of 4, with 3 observations?

Constructing Sampling Distributions Choose a sample size (n) Randomly select (independently) all possible combinations of n scores Compute the statistic of interest Plot the relative frequency of the values of the statistics computed Problem: with a population of 10 scores and an n of 3, there are 103 combinations (1000 independent random samples!)

Experiment 1 (The Smoking Study) How many cigarettes do you smoke per day?

Experiment (The Smoking Study) Score f p 1-10 19 0.03125 11-20 146 0.240131579 21-30 284 0.467105263 31-40 111 0.182565789 41-50 35 0.057565789 51-60 9 0.014802632 61-70 3 0.004934211 71-80 81-90 91-100 1 0.001644737 Total (n) 608 How many cigarettes do you smoke per day?

Experiment (The Smoking Study) How many cigarettes do you smoke per day? Postive skew Uni-modal asymetric

Experiment (The Smoking Study) How many cigarettes do you smoke per day? Mo = 20 Md = 20 M = 22.44 Range = 95 s2 = 97.33 s = 9.87

Experiment (The Smoking Study) How many cigarettes do you smoke per day? This is the probability distribution of SCORES.

Experiment (The Smoking Study) How many cigarettes do you smoke per day? Randomly select 10 ALL 608 Scores 10 20 34 23 15 18 40 27

Experiment (The Smoking Study) Return those 10 How many cigarettes do you smoke per day? ALL 608 Scores 10 20 34 23 15 18 40 27

Experiment (The Smoking Study) How many cigarettes do you smoke per day? Randomly select 10 more ALL 608 Scores 20 40 36 25 29 51 12 30 45

Experiment (The Smoking Study) Return those 10 How many cigarettes do you smoke per day? ALL 608 Scores 20 40 36 25 29 51 12 30 45

Experiment (The Smoking Study) How many cigarettes do you smoke per day? Randomly select 10 more ALL 608 Scores X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Experiment (The Smoking Study) Until Every Possible Combination has been Selected How many cigarettes do you smoke per day? ALL 608 Scores X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Experiment (The Smoking Study) Or 1,765,877,296,986,850,000,000 times How many cigarettes do you smoke per day? ALL 608 Scores X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Experiment (The Smoking Study) What if we do it 100 times? How many cigarettes do you smoke per day? ALL 608 Scores X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Experiment (The Smoking Study) What if we do it 100 times? How many cigarettes do you smoke per day?

Experiment (The Smoking Study) What if we do it 1000 times? How many cigarettes do you smoke per day?

Experiment (The Smoking Study) What if we do it ∞ times? How many cigarettes do you smoke per day?

Experiment (The Smoking Study) Sampling Distribution Of Sample Means Distribution of Scores Postive skew Uni-modal asymetric

Experiment (The Smoking Study) Sampling Distribution Of Sample Means Distribution of Scores Rectangular

Experiment (The Smoking Study) Sampling Distribution Of Sample Means Distribution of Scores Rectangular

Probability Theory to the Rescue The mean of the sampling distribution of the sample mean is equal to the population mean The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of n As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution (Central Limit Theorem) The Three Amazing Facts

Probability vs. Sampling Distributions All sampling distributions are probability distributions all the possible levels of a sample statistic and their probabilities But not all probability distributions are sampling distributions The probability distribution of scores is what a sampling distribution is constructed from

Central Limit Theorem (CLT) As sample size increases, the sampling distribution of the sample mean will approach a normal distribution The shape of the the distribution of scores DOES NOT MATTER - the sampling distribution of the sample mean will approach normality, as sample size increases

Usefulness of the CLT With large enough sample sizes, and the sampling distribution approaching normality, the probability of certain results can be found using z-score transformations and Table A

Example A Psychology class with 25 students had a mean SAT verbal score of 535 What is the probability of having a class of this size with a mean SATv of 535 or higher? We know that :

Example, cont.

Example (cont.) Area with M’s >535 z of 1.75 has a proportion of .9599 of the scores below it 95.99% of the means are at or below 535, we have a 1-.9599 = 4.01% chance of getting 535 or higher .9599 +1.75

Review of Concepts Random Sampling Sampling Distributions Central Limit Theorem Find the probability of obtaining some mean, with z-scores and table A