Statistics Chapter 6 - Lecture notes Lecture 6 PDF

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In-depth notes on sampling distribution and hypothesis testing including a question that will be on the exam as well as pictures in the notes ...

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Statistics Chapter 6 - Sampling Distribution  Exam question: When we are interested in drawing inferences from collected data, there are 3 frequency distributions of interest. Name and describe each of the 3 distributions. Describe the relationship between the distributions and how they are useful to inferential statistics ● From a sample we can calculate the mean and standard deviation ● Population also has mean and standard deviation ● We use samples to understand population ● The mean of the sample is based on sample size ● To compare M to the population directly it doesn't work ● z=x-μ ● The normal curve is a construct ● If the data is normal and symmetrical ○ We understand a lot about the nature of the curve itself ● All the possible M’s of a given size that could be derived from the population it came from  Frequency distributions of sample means ● A sample mean on average, is close to the mean of the population from which it came from ● Inferences about population parameters can be made from sample statistics ● Ex: if our assumptions about the data being normally distributed are valid, then the sample mean, on average, reflects the mean of the population ○ Some error - on average ● The means from different from samples obtained from the same underlying population are all estimates of μ ● What needs to be determined is the probability that a particular sample actually comes from a particular population ● Compare the sample mean that we have collected during the experiment to a distribution of all possible sample means from the population ● NOTE: this is not a comparison of our sample mean and the distribution of individual scores within the population ● There is always going to be some variability between the sample statistics we obtain from the experiment and the population parameter ○ Sampling error: the amount by which the sample estimate of the measure (say the mean or SD) differs from the underlying population parameter (μ or σ) ○ In general, the difference between the samples are simply due by chance

The Sampling Distribution ● Another theoretical distribution ● It is a distribution that gives the probability of all possible values of a sample mean ● When collecting sample statistics, we can exploit the distribution of means to male inferences about the population ● Ex: population scores: 2, 4, 6, 8. The μ = 5 and σ= 2.24 ○ We can create a distribution of all the possible random samples when n=2. We assume that sampling with replacement is happening ○ Takes 16 samples to get every combination

      ● Note that ○ The sample means tend to “pile up” around the population mean (recall from a previous slide that μ = 5) ○ The distribution is approximately normal in shape ● Another way of thinking about the distribution of means is that it is a probability distribution ● Mean of a sampling distribution is always the same ○ Ex: the probability of pulling a sample that has mean = 3 is:  ● We can calculate the parameter of the sampling distribution of the mean is:

Standard error of the mean ● The SEM is the standard amount of difference between a sample (M) and the population mean (μ). This represents the average amount of difference we can expect between the population mean and the sample mean ● The M and μ will usually be relatively close; however, there is always going to be some variability between these values for each sample. The σM quantifies the variability   ● Consider the original population of 4 scores, the μ=5 and the σ=2.24 σ2=5.01 ● Ex:

● Sample size increases, there is less variability (or less error) associated with predicting the population mean

● The central limit theorem: ○ Is a statistical principle that defines the μM, σM, and the shape of the theoretical sample distribution ○ In essence, it tells us that for a population with a given σ, μ, the distribution of sample means for sample size of n will approach a normal distribution with a mean of μ and a SD of σ divided by the square root of n ○ The distribution of sample means is the collection of sample means for all possible random samples of a given n that can be obtained from the population ○ A sampling distribution is a distribution of statistics obtained by selecting all of the possible samples of a specific size from a population ○ There are 2 fundamentally important facts that male the central limit theorem useful to us: ■ It describes the distribution of sample means for any population, irrespective of the population, shape, mean, or SD ■ The distribution of sample means “approaches” a normal distribution very quickly with increasing sample sizes (n)  Principles of central limit theorem ● The mean of the distribution of means is always equal to the mean of the population: ○ μM=μ ● The standard error measures the standard amount of difference that is expected simply due to chance :

● If the underlying population is normally distributed, then the distribution of means is also normally distributed ● If the underlying population distribution is not normal,  Ex: The population mean for the GRE is 558 with a standard deviation of 139. A researcher gives a training course to 100 candidates to see if their scores will improve. After the course, students take the GRE and score, on average, 585. Does the training course improve performance on the GRE? ● μ = 558 ● σ= 139 ● ●

N = 100 M = 585

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μM= 558 σM =13.9 z= 1.94 Sample that is less that 558 -> 50% Probability of getting 585 or less -> 0.9738 Mean greater than 585 -> p= 0.0262 Does the training course improve marks on the GRE ○ No Ex: a researcher undertakes a study of a treatment for depression. There are 120 depressed patients who are getting a new form of therapy ○ Population 1: all possible patients in existence that have met the criteria for depression and who are getting a new form of therapy ○ Population 2: all possible patients in existence that have met the criteria for depression and who do NOT receive the new therapy ○ The goal of the experiment is to see if after 4 weels of the new therapy, patients that have a new form of therapy have less depression than patients who do not get the new therapy Let's assume that the researcher is using a standardized test of depression like the Beck Depression Inventory or the Hamilton Depression Scale Since the DV is a standardized test, we know the statistical parameters of the underlying depressed population ○ μ = 69.5 ○ Σ = 14.1 Given the characteristics of the central limit theorem, we know the parameters of the distribution of means related to this population: ○ μM = 69.5 and

  ○ After 4 weeks of the new therapy, the subjects are all re-evaluated using the standard test that was used to make the initial diagnosis. The 120 subjects in the experiment are the sample. The mean depression score on the standardized test is 66.8. The million dollar question is “did the therapy make a difference”? ○ We need to determine how the sample mean compared to the population mean

○ The researcher has found the the treatment group has a score that is 2.09 SD’s below the mean of the depressed population. The researcher needs to decide if the treatment is effective is the treatment enough to make a clinical difference?  Hypothesis testing ● Means from different samples from the same population are estimates of μ ● The most important question when considering inferential statistics is “what is the probability that a particular sample mean comes from a particular population”? ● Compare the sample mean to a distribution of all possible sample means in the population and not to a distribution of individual scores from the population  Logic of hypothesis testing: ● We assume the experimental manipulation has no impact on the DV. that is, our manipulations make no difference ● If this basic assumption is wrong, then the conclusion that the manipulation does male a difference can be supported ● The confusion logic is necessary as mathematically, we generally only know about the circumstance where there are no differences ○ μM and σM ● Ex: a person claims that she can identify people who have above average intelligence with their eyes closed. We take her to a full stadium and blindfold her. We ask her to pick out someone with greater than average IQ ○ We know that IQ is normally distributed in the population with μ = 100 and σ = 15 ○ We know that if she picks out someone with an IQ of 145 (3 DS above the mean), it is unlikely that we can attribute her performance to chance ○ Since 145 is 3 SD’s aboce the mean, we know that the relevant z-score is +3.0 ○ Is we consult the tables, we find that the probability of picking someone with an IQ of 130 as only 2.3% of scores are higher than that ○ What if she selects someone with an IQ of 115? Are we as impressed? There is a probability of picking this score or higher equal to 15.9% ■ We might consider this an inconclusive result

○ The deal with the inconclusive, we might decide a printer on an IQ score that, if she successfully picked a person that has that IQ, we would be impressed by ■ We could say that we want her to pick people out with a 2% probability of being wrong. That would mean that we said “ pick out people with IQ’s greater than 130  The Jargon ● Null hypothesis ○ H0 - a statement that suggest that the treatment of interests will have NO outcome on the results of the experiment ○ The H0 acknowledges that we are assuming that the experiment will not describe or influence an existing natural phenomenon ● The alternative hypothesis ○ H1, ○ Logically, the complete opposite to the H0 ○ The H0 and H1, must be mutually exclusive and exhaustive  Steps in hypothesis testing ● Ex: we are researchers who want to investigate the effectiveness of a new stress reduction method that we have developed: ○ Suppose a subject is randomly selected from the general population. The subject is treated with the new stress reduction method. Then, his or her stress is measured ○ We will assume that we know the population parameters related to stress within the population from which the subject was taken: ■ μ= 40, σ= 10 ● We can reframe the research question into null and alternative hypotheses about the specific population that we are interested in: ○ Population 1: all the people in the world who have not been trained in the new method ○ Population 2: all people in the world who have been trained in the stress reduction method ■ H0: Those trained in the method do not have a lower stress score ■ H1: those trained in the new method do have lower stress scores ● We now need to think about the information that we do not have about stress in the general population ○ μ= 40, σ= 10

○ We know absolutely nothing about the other population (population 2) other than the stress levels of the subjects selected from population 1 ○ Does the performance of the subject most probably represents the general population (population 1 as represented by the H0) or is their score so extreme that it is unlikely to represent the general population and therefore, most likely represents another totally different population - the population of all possible people who received the new stress reducing treatment (population 2 represented by H1) ● Decide the point at which we are comfortable with saying “no that score is too extreme from the general population” ○ Establish the “cutoff” sample score at which the H0 should be rejected in favour of the H1 ○ Accept a 1% possibility of making a mistake in declaring our new treatment a success ○ Another way of saying this is that for our treatment “successful” the subjects has to have a score that has the probability of being from the general population of 0.01 or less ■ In the jargon, we are setting a=0.01 ○ We will need to know the point on the distribution that will “cut off” the lower 1% of the distribution ○ Consulting the z score tables, the z score that is consistent with 1% is ± 2.33

● Suppose that the subject scored 15 on the standardized stress scale that we know the population parameters for ● Z score for this raw datum

● Contrast the subjects score with the general population to see if the new treatment caused the subject to experience decreased amounts of stress such that their experimental score is unlikely to represent the general population ○ Calculated that the subjects performance yielded a z score of -2.5. This is more extreme than our cut off point

○ Therefore, we reject H0 that there is no difference between the subjects score and the scores of the general population ○ Since the H0 is rejected, the H1, must be supported. We conclude that the subjects score is unlikely to have come from the population of people who did not get the new treatment ○ We further conclude that we are accepting a 1% chance of being wrong in our determination ○ The most interesting conclusion is that our new stress reduction method leads to lower stress scores (p...