The Sampling Distribution Introduction to Hypothesis Testing and Interval Estimation

Outline  Distinctions  Sampling Distribution  The Central Limit Theorem  Confidence Intervals

Random Sampling

Key things to keep in mind  Population- what we want to talk about  Sample- what we have with our data  Sampling distribution- the means by which we will go from our sample to the population

Sampling Distribution  Sampling distributions concern any statistic we can come up with. Examples: Measures of Central Tendency Measures of Variability Measures of Relationship Ratios  Sample != sampling distribution  Recall also that sampling distributions can be theoretical (used in most studies) or empirical (seeing wider use via bootstrapping).  It is the properties of the sampling distribution

Central Limit Theorem (CLT)  Suppose X is random mean  standard deviation not necessarily normal

Terms Concerning Sampling Distribution of the Mean  Standard Error of the mean: Is just the standard deviation of the sampling distribution.  i.e. it is a particular standard deviation  Sampling error The sample cannot be fully representative of the population As such, there is variability due to chance We could have a thousand sample means and none of them equal exactly the population mean. However…

CLT (continued)  Properties of the sampling distribution of the mean random has a mean of  has a standard error Distributed approximately normal for large samples Normal for all samples if the variable X is normal

The Central Limit Theorem  For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution as the sample size (N) gets larger. This of course begs the question of what is ‘large enough’  Furthermore, the sampling distribution of the mean will have a mean equal to µ (the population mean), and a standard deviation equal to

Central Limit Theorem  With the mean, we can use sample data and the normal curve to reach conclusions about the population of interest  We of course desire large, random samples in order to do Non-random selection can result in under-selection or over-selection of subsections of the population.  e.g. carry out a telephone opinion poll  ampling_dist/index.html ampling_dist/index.html

In summary: sample means  are random  are normally distributed for large sample sizes  distribution has mean   distribution has standard error (standard deviation)

Confidence intervals  Draw a sample, gives us a mean  is our best guess at µ  For most samples will be close to µ  is a ‘point’ estimate  However, we can also give a range or interval estimate that takes into account the uncertainty involved in that estimate Using the normal distribution

Confidence interval equation Where = sample mean Z = z value from normal curve = standard error of the mean

95% confidence interval  Let’s say we want a 95% confidence interval.  Obtain 1 the ‘critical’ z-score for p = % above +z, and 2.5% below -z  p =.025 then z = 1.96  When the population standard deviation is not known, we use the t critical value instead

Confidence interval example  Randomly selected a group of 10 of you folks with a mean score of 89 (s = 6) on the midterm.  What guess can we make as to the true mean of the class?

 *  (1.90)  ( ) < < ( )  < <  This seems pretty wide; it essentially covers a full letter grade. Why do you think that is?

Important: what a confidence interval means  A 95% confidence interval means that: 95% of the confidence intervals calculated on repeated sampling of the same population will contain µ  Note that the population value does not vary i.e. it’s not a 95% chance that it falls in that specific interval 1  In other words, the CI attempts to capture the true population mean, but we would have a different interval estimate for each sample drawn  x.html x.html  In R library(animation) conf.int(.95)

Question to think about  How does one know if the confidence interval calculated actually contains the true population mean?