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Understand t-tests, CLT, & variance rules applied to means – both known and unknown. Learn how to analyze samples to derive accurate conclusions.
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HYPOTHESIS TESTING APPLIED TO MEANS Chapter 7
Outline • Central Limit Theorem • Single Means – σ is known • Single Means – σ is unknown • Pairs of Means – Matched Samples • Pairs of Means – Independent Samples • Variance Sum Law • Pooling Variances & Unequal Ns • Heterogeneity of Variance • The Cookbook Chapter 7
Typical Question Q1: Is some sample mean different from what would be expected given some population distribution? • On the face of it, this question should remind you of your previous fun with z-scores. • In the case of z-scores, we asked whether some observation was significantly different from some sample mean. • In the case of this question, we are asking whether some sample mean is significantly different from some population mean Chapter 7
Typical Question • Despite this apparent similarity, the questions are different because the sampling distribution of the mean (the t distribution) is different from the sampling distribution of observations the z distribution). • In order to understand the distinction between the z and t-tests, we need to understand the Central Limit Theorem. . . Chapter 7
Central Limit Theorem • CLT: Given a population with mean μ and variance σ, the sampling distribution of the mean (the distribution of sample means) will have a mean equal to μ (i.e., ), a variance ( ) equal to , and a standard deviation ( ) equal to . The distribution will approach the normal distribution as N, the sample size, increases. Chapter 7
Central Limit Theorem • Population of Scores 1, 2, 4, 5, 8 N = 5 μx = 4.00 σX = 2.45 Chapter 7
Central Limit Theorem • All possible samples of size 2 (n=2) 1,1 2,1 4,1 5,1 8,1 1,2 2,2 4,2 5,2 8,2 1,4 2,4 4,4 5,4 8,4 1,5 2,5 4,5 5,5 8,5 1,8 2,8 4,8 5,8 8,8 Chapter 7
Central Limit Theorem 3. Sampling distribution of sample means 1.0 1.5 2.5 3.0 4.5 1.5 2.0 3.0 3.5 5.0 2.5 3.0 4.0 4.5 6.0 3.0 3.5 4.5 5.0 6.5 4.5 5.0 6.0 6.5 8.0 Chapter 7
Central Limit Theorem 4. Chapter 7
Central Limit Theorem 5. Chapter 7
Single Means – σ is known • Although it is seldom the case, sometimes we know the variance (as well as the mean) of the population distribution of interest. • In such cases, we can do a revised version of the z-test that takes into account the central limit theorem Chapter 7
Single Means – σ is known • Specifically... ...becomes... Chapter 7
Single Means – σ is known • With this formula, we can answer questions like the following: Say I sampled 25 students at UofT and measured their IQ, finding a mean of 110. Is this mean significantly different from the population which has a mean IQ of 100 and a standard deviation of 15? Chapter 7
Single Means – σ is unknown • Unfortunately, it is very rare that we know the population standard deviation. • Instead we must use the sample standard deviation, s, to estimate . • However, there is a hitch to this. While s2 is an unbiased estimator of 2 (i.e., the mean of the sampling distribution of s2 equals 2), the sampling distribution of s2 is positively skewed Chapter 7
Single Means – σ is unknown • This means that any individual s2 chosen from the sampling distribution of s2 will tend to underestimate 2. • Thus, if we used the formula that we used when was known, we would tend to get z values that were larger than they should be, leading to too many significant results Chapter 7
Single Means – σ is unknown • The solution? Use the same formula (modified to use s instead of ), find its distribution under H0, then use that distribution for doing hypothesis testing. The result: Chapter 7
Single Means – σ is unknown • When a t-value is calculated in this manner, it is evaluated using the t-table (p. 747 of the text) and the row for n-1 degrees of freedom. • So, with all this in hand, we can now answer questions of the following type... Chapter 7
Single Means – σ is unknown Example: Let’s say that the average human who has reached maturity is 68” tall. I’m curious whether the average height of our class differs from this population mean. So, I measure the height of the 100 people who come to class one day, and get a mean 70” and a standard deviation of 5”. What can I conclude? Chapter 7
Single Means – σ is unknown • If we look at the t-table, we find the critical t-value for alpha=.05 and 99 (n-1) degrees of freedom is 1.984. • Since the tobt > tcrit, we reject H0 Chapter 7
Typical Questions • Quite often, instead of comparing a single score or a single mean to a population, we want to compare two means against one another • In other words – is the mean of one group significantly different from another? • Matched Sample • Independent Sample Chapter 7
Pairs of Means – Matched Samples • In many studies, we test the same subject on multiple sessions or in different test conditions. • We then wish to compare the means across these sessions or test conditions. • This type of situation is referred to as a pair wise or matched samples (or within subjects) design, and it must be used anytime different data points cannot be assumed to be independent • sexist profs example Chapter 7
Pairs of Means –Matched Samples • As you are about to see, the t-test used in this situation is basically identical to the t-test discussed in the previous section, once the data has been transformed to provide difference scores Chapter 7
Pairs of Means –Matched Samples • Assume we have some measure of rudeness and we then measure 10 profs rudeness index; once when the offending TA is male, and once when they are female. Chapter 7
Pairs of Means –Matched Samples • Question becomes, is the average difference score significantly different from 0? • So, when we do the math: Chapter 7
Pairs of Means –Matched Samples • The critical t with alpha equal .05 (two-tailed) and 9 (n-1) degrees of freedom is 2.262. • Since tobt is not greater than tcrit, we can not reject H0. • Thus, we have no evidence that the profs rudeness is difference across TAs of different genders Chapter 7
Pairs of Means –Independent Samples • Another common situation is one where we have two of more groups composed of independent observations. • That is, each subject is in only one group and there is no reason to believe that knowing about one subjects performance in one of the groups would tell you anything about another subjects performance in one of the other groups Chapter 7
Pairs of Means –Independent Samples • In this situation we are said to have independent samples or, as it is sometimes called, a between subjects design • Example: Let’s take the famous “Misinformation Effect” memory experiment where subjects see a video of a car accident and are asked to estimate the speed of the car involved in the accident. The adjective used to describe the collision (smashed vs. ran into vs. contacted) is varied across groups (n=20). Did the manipulation affect speed estimates? That is, are the mean speed estimates of the various groups different? Chapter 7
Pairs of Means –Independent Samples • In the accident video, about how fast (in km/h) do you think the gray car was going when it ________ the side of the red car? Chapter 7
Pairs of Means –Independent Samples • There are, in fact, three different t-tests we can perform in this situation, comparing groups 1 &2, 1&3, or 2&3. • For demonstration purposes, let’s only worry about groups 1 & 2 for now. • So, we could ask, do subjects in Group 1 give different estimates of the gray car’s speed than subjects in Group 2? Chapter 7
Variance Sum Law • When testing a difference between two independent means, we must once again think about the sampling distribution associated with H0. • If we assume the means come from separate populations, we could simultaneously draw samples from each population and calculate the mean of each sample Chapter 7
Variance Sum Law • If we repeat this process a number of times, we could generate sampling distributions of the mean of each population, and a sampling distribution of the difference of the two means. • If we actually did this, we would find that the sampling distribution of the difference would have a variance equal to the sum of the two population variances Chapter 7
Variance Sum Law • In fact, the Variance Sum Law states that: The variance of a sum or difference of two independent variables is equal to the sum of their variances Chapter 7
Pairs of Means –Independent Samples • Now recall that when we performed a t-test in the situation where the population standard deviation was unknown, we used the formula: • Given all of the above, we can now alter this formula in a way that will allow us to use it in the independent means example Chapter 7
Pairs of Means –Independent Samples • Specifically, instead of comparing a single sample mean with some mean, we want to see if the difference between two sample means equals zero. • Thus the numerator (top part) will change to: or simply Chapter 7
Pairs of Means –Independent Samples • And, because the standard error associated with the difference between two means is the sum of each mean’s standard error (by the variance sum law), the denominator of the formula changes to Chapter 7
Pairs of Means –Independent Samples • Thus, the basic formula for calculating a t-test for independent samples is: Chapter 7
Pairs of Means –Independent Samples • Finishing the example: Comparing groups 1 and 2, we end up with: • df = (n1+n2-2) = 38, tCRIT = 2.021. • Since tOBT > tCRIT we reject H0 Chapter 7
Pooling Variances & Unequal Ns • The previous formula is fine when sample sizes are equal. • However, when sample sizes are unequal, it treats both of the S2 as equal in terms of their ability to estimate the population variance Chapter 7
Pooling Variances & Unequal Ns • Instead, it would be better to combine the s2 in a way that weighted them according to their respective sample sizes. This is done using the following pooled variance estimate: Chapter 7
Pooling Variances & Unequal Ns • Given this, the new formula for calculating an independent groups t-test is: Chapter 7
Pooling Variances & Unequal Ns • Using the pooled variances version of the t formula for independent samples is no different from using the separate variances version when sample sizes are equal. It can have a big effect, however, when sample sizes are unequal. Chapter 7
Heterogeneity of Variance • The text book has a large section on heterogeneity of variance (pp 213-216) including lots of nasty looking formulae. All I want you to know is the following: • When doing a t-test across two groups, you are assuming that the variances of the two groups are approximately equal. • If the variances look fairly different, there are tests that can be used to see if the difference is so great as to be a problem Chapter 7
Heterogeneity of Variance • If the variances are different across the groups, there are ways of correcting the t-test to take the heterogeneity in account. • In fact, t-tests are often quite robust to this problem, so you don’t have to worry about it too much. Chapter 7
The Cookbook One Observation vs. Population : Chapter 7
The Cookbook One Mean vs. One Population Mean: Population variance known: Chapter 7
The Cookbook One Mean vs. One Population Mean: Population variance unknown: df = n-1 Chapter 7
The Cookbook Two Means: Matched samples: first create a difference score, then... df = nD-1 Chapter 7
The Cookbook Two Means: Independent samples: df = n1+n2-2 Chapter 7
The Cookbook Two Means: Independent samples. . .continued: where: Easy as baking a cake, right? Now for some examples of using these recipes to cook up some tasty conclusions. . . Chapter 7
Examples 1) The population spends an average of 8 hours per day working, with a standard deviation of 1 hour. A certain researcher believes that profs work less hours than average and wants to test whether the average hours per day that profs work is different from the population. This researcher samples 10 professors and asks them how many hours they work per day, leading to the following data set: • perform the appropriate statistical test and state your conclusions. Chapter 7
