Education 793 Class Notes Inference and Hypothesis Testing Using the Normal Distribution 8 October 2003

2 Today’s agenda Class and lab announcements What questions do you have? Inference and hypothesis testing

3 Inferential statistics Statistics that allow one to draw conclusions from data (as opposed to simply describing data) Characteristics of samples are described by statistics; characteristics of populations are described by parameters Using probability and information about a sample to draw conclusions (make inferences) about a population or how likely it is that a result could have been obtained simply as a function of chance

4 Chain of reasoning Random selection can take several forms (such as simple, systematic, cluster, or stratified random sampling), and is intended to generate a sample that represents the population.

5 Three distributions of note Distribution in the population (parameters) Distribution in the sample (statistics) Distribution of sample statistics from all possible samples of a given size drawn from the population, or the sampling distribution of a statistic (mean, correlation coefficient, regression coefficient, etc.)

6 Comparing the three distributions PopulationSampleSample means Mean =  Mean = Mean =  Stddev =  Stddev = s Stddev =  Population distribution Sample distribution Sampling distribution of the mean

7 Hypothesis testing Used to assess the statistical significance of findings, and involves the comparison of empirically observed findings (drawn from a sample) with those which were expected. Null hypothesis (H 0 ): The hypothesis about the population, suggesting no difference (hence null), such as: 1) a distribution has a specified mean 2) two or more statistics (such as means) are not the same Research hypothesis (H A ): Any hypothesis that is an alternative to the one being tested; usually the opposite of the null hypothesis. By rejecting the null hypothesis one shows that there is evidence to suggest that the research hypothesis may be true.

8 Null and Alternative Hypotheses Examples, One population –Null:  =some specific value –Alternative:  some specific value (nondirectional - 2-tailed) OR  < some specific value (directional – 1-tailed) OR  > some specific value (directional – 1-tailed) Examples, Two populations –Null:  1=  2 –Alternative  1  2 (nondirectional) OR  1 >  2 (directional) OR  1 <  2 (directional)

9 Inferential error Type I: Alpha –Rejecting the null hypothesis with the null hypothesis is really true Type II: Beta –Failing to reject the null hypothesis when the null hypothesis is in fact false

10 Possible outcomes Decision In the population, H 0 is true In the population, H 0 is false Reject H 0 (H 0 false) Type I error  Correct decision Fail to reject H 0 (H 0 true) Correct decision Type II error 

11 Basic Concepts of Hypothesis Testing The null hypothesis is assumed to be true The greater the difference between the sample mean and hypothesized mean, the lower the probability that the difference is due to chance Need to understand sampling distribution of the mean

12 Dancing distributions X freq Sample 1 X freq Sample 2 X freq Sample n X 2 X n X 1 Population X freq Mu X

13 Sampling distribution of the mean Shape of this distribution is defined by the Central Limit Theorem: As sample size increases the sampling distribution of the mean approximates a normal distribution. Central tendency: The mean of the distribution is an unbiased estimator of the population parameter value (  ). Variability: The variance of the distribution is a function of the variance of the population (  ) and the size of the sample.

14 Central Limit Theorem For a given a distribution with a mean  and variance  2, the sampling distribution of the mean approaches a normal distribution with a mean (  ) and a variance  2 /N as N, the sample size, increases. The sampling distribution of the mean approaches a normal distribution regardless of the shape of the original distribution. Really, it does! Hard to believe, but true!! simulation

15 Which sample size? According to the Central Limit Theorem, as the sample size increases, the sampling distribution: A. becomes more normal in shape B. decreases in variability (less spread) In this case, sample size refers to the size of each individual sample, not the number of samples drawn (which, in practice, is typically one). To generate a complete sampling distribution of the mean requires drawing an infinite number of samples of a particular sample size, and no one – literally – has time for that!

16 Three population distributions Each distribution ranges from 0 to 100, with a of 50 freq

17 Central limit: Normal population

18 Central limit: Skewed population

19 Central limit: Rectangular population

20 How can we use this information? We can draw a sample from a population, and generate statistics that describe that particular sample. If the sample is of sufficient size, based on the central limit theorem, we can make inferences about unmeasured characteristics of the population (parameters, such as  and  2 ) based on characteristics of the sample (statistics, such a mean and variance). Since the sampling distribution of the mean approaches a standard normal distribution as the sample size increases (regardless of the shape of the original distribution), with a sufficient sample we can assign probability statements to the inferences described above.

21 Point Estimates Most statistics are point estimates -- the single best guess ( ) of the population parameter:

22 Standard deviation or standard error? A standard deviation is a measure of the variability in a distribution of scores. A standard error is a measure of sampling error, and is the standard deviation of the sampling distribution of a statistic (such as a mean). We seldom have a direct measure of the standard error of a statistic, so it must be estimated. For means, standard error can be estimated in two ways. – If the population standard deviation is known: –I f not, standard error can be estimated from the sample’s standard deviation, s:

23 Hypothesis Testing Process Compute test statistic from sample data Identify sampling distribution of statistic (assuming nothing unusual is happening). This becomes the null hypothesis – necessary for testing purposes, but not typically the research hypothesis in which we are interested. Compare computed statistic to sampling distribution based upon null and research hypotheses (e.g., one or two tailed test? Need to reduce erroneous conclusions?) Make inference – If the value of the test statistic is unusual (lies in the critical region), reject the null hypothesis and infer that something besides expected variation is going on.

24 A sample problem A random sample of 36 teachers scores an average of 1100 on a national teacher assessment test. The test is designed to produce a standard normal distribution, with a mean of 950 and variance of 250,000. – A. Using alpha =.05, test the hypothesis that the mean teachers’ score was significantly higher than the national average. =1.8  = CV 1.8 Reject Null

25 Problem to Try 49 seniors scored a mean of 54 on a national chemistry exam. Using alpha=.05, test the hypothesis that the seniors were below the national norm of 58 with an SD of 14.

26 Next week Chapter 10 p