Statistics for the Social Sciences Psychology 340 Fall 2006 Sampling distribution

Outline Review 138 stuff: What are sample distributions Central limit theorem Standard error (and estimates of) Test statistic distributions as transformations

Based on standard error or an estimate of the standard error Testing Hypotheses From last time: Core logic of hypothesis testing Considers the probability that the result of a study could have come about if the experimental procedure had no effect If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported How do we determine this? Based on standard error or an estimate of the standard error

Flipping a coin example Number of heads HHH 3 HHT 2 HTH 2 HTT 1 2 THH THT 1 TTH 1 TTT 2n = 23 = 8 total outcomes

Flipping a coin example Number of heads 3 Distribution of possible outcomes (n = 3 flips) 2 X f p 3 1 .125 2 .375 Number of heads 1 2 3 .1 .2 .3 .4 probability 2 .375 .375 1 2 .125 .125 1 1

Hypothesis testing Distribution of Sample Means Distribution of possible outcomes (of a particular sample size, n) Can make predictions about likelihood of outcomes based on this distribution. In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions) This distribution of possible outcomes is often Normally Distributed

Hypothesis testing Distribution of Sample Means Distribution of possible outcomes (of a particular sample size, n) Mean of a group of scores Comparison distribution is distribution of means

Distribution of sample means Distribution of sample means is a “theoretical” distribution between the sample and population Mean of a group of scores Comparison distribution is distribution of means Population Distribution of sample means Sample

Distribution of sample means A simple case Population: 2 4 6 8 All possible samples of size n = 2 Assumption: sampling with replacement

Distribution of sample means A simpler case Population: 2 4 6 8 All possible samples of size n = 2 There are 16 of them mean 2 2 2 6 4 5 6 4 7 8 4 6 8 2 2 4 3 8 4 6 2 4 6 4 8 2 8 2 5 4 2 3 4 4

Distribution of sample means 2 3 4 5 6 7 8 1 In long run, the random selection of tiles leads to a predictable pattern mean mean mean 2 2 2 4 6 5 8 2 5 2 4 3 4 8 6 8 4 6 2 6 4 6 2 4 8 6 7 2 8 5 6 4 5 8 8 8 4 2 3 6 6 6 4 4 4 6 8 7

Distribution of sample means 2 3 4 5 6 7 8 1 Sample problem: What’s the probability of getting a sample with a mean of 6 or more? X f p 8 1 0.0625 7 2 0.1250 6 3 0.1875 5 4 0.2500 P(X > 6) = .1875 + .1250 + .0625 = 0.375 Same as before, except now we’re asking about sample means rather than single scores

Properties of the distribution of sample means Shape If population is Normal, then the dist of sample means will be Normal If the sample size is large (n > 30), regardless of shape of the population Distribution of sample means Population N > 30

Properties of the distribution of sample means Center The mean of the dist of sample means is equal to the mean of the population Population Distribution of sample means same numeric value different conceptual values

Properties of the distribution of sample means Center The mean of the dist of sample means is equal to the mean of the population Consider our earlier example 2 4 6 8 Population Distribution of sample means means 2 3 4 5 6 7 8 1 2 + 4 + 6 + 8 4 m = = 5 2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8 16 = = 5

Properties of the distribution of sample means Spread The standard deviation of the distribution of sample mean depends on two things Standard deviation of the population Sample size

Properties of the distribution of sample means Spread Standard deviation of the population The smaller the population variability, the closer the sample means are to the population mean m X 1 2 3 m X 1 2 3

Properties of the distribution of sample means Spread Sample size n = 1 m X

Properties of the distribution of sample means Spread Sample size n = 10 m X

Properties of the distribution of sample means Spread Sample size m n = 100 The larger the sample size the smaller the spread X

Properties of the distribution of sample means Spread Standard deviation of the population Sample size Putting them together we get the standard deviation of the distribution of sample means Commonly called the standard error

Standard error The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean In other words, it is an estimate of the error that you’d expect by chance (or by sampling)

Distribution of sample means Keep your distributions straight by taking care with your notation Population s m Distribution of sample means Sample s X

Properties of the distribution of sample means All three of these properties are combined to form the Central Limit Theorem For any population with mean  and standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of  and a standard deviation of as n approaches infinity (good approximation if n > 30).

Performing your statistical test What are we doing when we test the hypotheses? Computing a test statistic: Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error

Hypothesis Testing With a Distribution of Means It is the comparison distribution when a sample has more than one individual Find a Z score of your sample’s mean on a distribution of means

“Generic” statistical test An example: One sample z-test Memory example experiment: Step 1: State your hypotheses We give a n = 16 memory patients a memory improvement treatment. H0: the memory treatment sample are the same (or worse) as the population of memory patients. After the treatment they have an average score of = 55 memory errors. mTreatment > mpop > 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? HA: Their memory is better than the population of memory patients mTreatment < mpop < 60

“Generic” statistical test An example: One sample z-test Memory example experiment: H0: mTreatment > mpop > 60 HA: mTreatment < mpop < 60 We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria One -tailed After the treatment they have an average score of = 55 memory errors. a = 0.05 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8?

“Generic” statistical test An example: One sample z-test Memory example experiment: H0: mTreatment > mpop > 60 HA: mTreatment < mpop < 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed a = 0.05 Step 3: Collect your data After the treatment they have an average score of = 55 memory errors. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8?

“Generic” statistical test An example: One sample z-test Memory example experiment: H0: mTreatment > mpop > 60 HA: mTreatment < mpop < 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed a = 0.05 Step 4: Compute your test statistics After the treatment they have an average score of = 55 memory errors. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? = -2.5

“Generic” statistical test An example: One sample z-test Memory example experiment: H0: mTreatment > mpop > 60 HA: mTreatment < mpop < 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed a = 0.05 After the treatment they have an average score of = 55 memory errors. Step 5: Make a decision about your null hypothesis How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? 5% Reject H0

“Generic” statistical test An example: One sample z-test Memory example experiment: H0: mTreatment > mpop > 60 HA: mTreatment < mpop < 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed a = 0.05 After the treatment they have an average score of = 55 memory errors. Step 5: Make a decision about your null hypothesis - Reject H0 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? - Support for our HA, the evidence suggests that the treatment decreases the number of memory errors