Psy B07 Chapter 7Slide 1 HYPOTHESIS TESTING APPLIED TO MEANS

Chapter 7Chapter 7 Slide Slide 11

Psy B07

HYPOTHESIS TESTING HYPOTHESIS TESTING APPLIED TO MEANSAPPLIED TO MEANS


Psy B07

Central Limit TheoremCentral Limit Theorem Single Means – Single Means – σσ is known is known Single Means – Single Means – σσ is unknown is unknown Pairs of Means – Matched SamplesPairs of Means – Matched Samples Pairs of Means – Independent SamplesPairs of Means – Independent Samples Variance Sum LawVariance Sum Law Pooling Variances & Unequal NsPooling Variances & Unequal Ns Heterogeneity of VarianceHeterogeneity of Variance The CookbookThe Cookbook

OutlineOutline


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Typical QuestionTypical Question

Q1Q1: Is some sample mean different from what : Is some sample mean different from what would be expected given some population would be expected given some population distribution?distribution?

On the face of it, this question should remind On the face of it, this question should remind you of your previous fun with z-scores.you of your previous fun with z-scores. In the case of z-scores, we asked whether In the case of z-scores, we asked whether

some some observationobservation was significantly different was significantly different from some from some sample meansample mean..

In the case of this question, we are asking In the case of this question, we are asking whether some whether some sample meansample mean is significantly is significantly different from somedifferent from some population mean population mean


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Typical QuestionTypical Question

Despite this apparent similarity, the Despite this apparent similarity, the questions are different because the questions are different because the sampling distribution of the mean (the t sampling distribution of the mean (the t distribution) is different from the distribution) is different from the sampling distribution of observations the sampling distribution of observations the z distribution).z distribution).

In order to understand the distinction In order to understand the distinction between the z and t-tests, we need to between the z and t-tests, we need to understand the Central Limit Theorem. . . understand the Central Limit Theorem. . .


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Central Limit TheoremCentral Limit Theorem

CLT: Given a population with mean CLT: Given a population with mean μμ and and

variance variance σσ, the sampling distribution of , the sampling distribution of

the mean (the distribution of sample the mean (the distribution of sample

means) will have a mean equal to means) will have a mean equal to μμ (i.e., (i.e.,

), a variance ( ) equal to , ), a variance ( ) equal to ,

and a standard deviation ( ) equal to and a standard deviation ( ) equal to

. The distribution will approach the . The distribution will approach the

normal distribution as N, the sample size, normal distribution as N, the sample size,

increases.increases.

x2

x n2

x n


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1.1. Population of ScoresPopulation of Scores

1, 2, 4, 5, 81, 2, 4, 5, 8

N = 5N = 5

μμxx = 4.00 = 4.00

σσXX = 2.45 = 2.45


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2.2. All possible samples of size 2 All possible samples of size 2 (n=2)(n=2)

1,11,1 2,12,1 4,14,1 5,15,1 8,1 8,1

1,21,2 2,22,2 4,24,2 5,25,2 8,28,2

1,41,4 2,42,4 4,44,4 5,45,4 8,48,4

1,51,5 2,52,5 4,54,5 5,55,5 8,58,5

1,81,8 2,82,8 4,84,8 5,85,8 8,88,8


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3. Sampling distribution of sample 3. Sampling distribution of sample meansmeans

1.01.0 1.51.5 2.52.5 3.03.0 4.54.5

1.51.5 2.02.0 3.03.0 3.53.5 5.05.0

2.52.5 3.03.0 4.04.0 4.54.5 6.06.0

3.03.0 3.53.5 4.54.5 5.05.0 6.56.5

4.54.5 5.05.0 6.06.0 6.56.5 8.08.0


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4. 4.

00.4N

x

x

x

)( x


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5.5.

73.12

45.2

n

73.1N

)x(

xx

x

2

xx


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Single Means – Single Means – σσ is known is known

Although it is seldom the case, Although it is seldom the case, sometimes we know the variance sometimes we know the variance (as well as the mean) of the (as well as the mean) of the population distribution of interest.population distribution of interest.

In such cases, we can do a revised In such cases, we can do a revised version of the z-test that takes into version of the z-test that takes into account the central limit theoremaccount the central limit theorem


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Specifically...Specifically...

...becomes......becomes...

x

z

n

xxz

x


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With this formula, we can answer questions like With this formula, we can answer questions like the following: Say I sampled 25 students at UofT the following: Say I sampled 25 students at UofT and measured their IQ, finding a mean of 110. Is and measured their IQ, finding a mean of 110. Is this mean significantly different from the this mean significantly different from the population which has a mean IQ of 100 and a population which has a mean IQ of 100 and a standard deviation of 15?standard deviation of 15?

33.3

25

15100110

n

xz


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Single Means – Single Means – σσ is is unknownunknown

Unfortunately, it is very rare that we Unfortunately, it is very rare that we know the population standard deviation.know the population standard deviation.

Instead we must use the sample standard Instead we must use the sample standard deviation, s, to estimate deviation, s, to estimate ..

However, there is a hitch to this. While However, there is a hitch to this. While ss22 is an unbiased estimator of is an unbiased estimator of 22 (i.e., the (i.e., the mean of the sampling distribution of smean of the sampling distribution of s22 equals equals 22), the sampling distribution of s), the sampling distribution of s22 is positively skewedis positively skewed


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This means that any individual sThis means that any individual s22 chosen from the sampling distribution chosen from the sampling distribution of sof s22 will tend to underestimate will tend to underestimate 22..

Thus, if we used the formula that we Thus, if we used the formula that we used when used when was known, we would was known, we would tend to get z values that were larger tend to get z values that were larger than they should be, leading to too than they should be, leading to too many significant resultsmany significant results


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The solution? Use the same formula The solution? Use the same formula (modified to use s instead of ), find its (modified to use s instead of ), find its distribution under Hdistribution under H00, then use that , then use that distribution for doing hypothesis testing.distribution for doing hypothesis testing.

The result:The result:

n

sx

t


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When a t-value is calculated in this When a t-value is calculated in this manner, it is evaluated using the t-manner, it is evaluated using the t-table (p. 747 of the text) and the table (p. 747 of the text) and the row for n-1 degrees of freedom.row for n-1 degrees of freedom.

So, with all this in hand, we can So, with all this in hand, we can now answer questions of the now answer questions of the following type...following type...


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Example:Example:

Let’s say that the average human who has Let’s say that the average human who has reached maturity is 68” tall. I’m curious whether reached maturity is 68” tall. I’m curious whether the average height of our class differs from this the average height of our class differs from this population mean. So, I measure the height of the population mean. So, I measure the height of the 100 people who come to class one day, and get a 100 people who come to class one day, and get a mean 70” and a standard deviation of 5”. What mean 70” and a standard deviation of 5”. What can I conclude?can I conclude?

45.0

2

100

56870

n

sx

t


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If we look at the t-table, we find the If we look at the t-table, we find the critical t-value for alpha=.05 and 99 critical t-value for alpha=.05 and 99 (n-1) degrees of freedom is 1.984. (n-1) degrees of freedom is 1.984.

Since the tSince the tobtobt > t > tcritcrit, we reject H, we reject H00


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Typical QuestionsTypical Questions

Quite often, instead of comparing a Quite often, instead of comparing a single score or a single mean to a single score or a single mean to a population, we want to compare population, we want to compare two means against one anothertwo means against one another

In other words – is the mean of one In other words – is the mean of one group significantly different from group significantly different from another?another?

Matched SampleMatched Sample Independent SampleIndependent Sample


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Pairs of Means – Pairs of Means – Matched SamplesMatched Samples

In many studies, we test the same In many studies, we test the same subject on multiple sessions or in subject on multiple sessions or in different test conditions.different test conditions.

We then wish to compare the means We then wish to compare the means across these sessions or test conditions.across these sessions or test conditions.

This type of situation is referred to as a This type of situation is referred to as a pair wise or matched samples (or within pair wise or matched samples (or within subjects) design, and it must be used subjects) design, and it must be used anytime different data points cannot be anytime different data points cannot be assumed to be independent assumed to be independent sexist profs examplesexist profs example


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Pairs of Means –Pairs of Means –Matched SamplesMatched Samples

As you are about to see, the t-test As you are about to see, the t-test used in this situation is basically used in this situation is basically identical to the t-test discussed in identical to the t-test discussed in the previous section, once the data the previous section, once the data has been transformed to provide has been transformed to provide difference scoresdifference scores


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Assume we have Assume we have

some measure of some measure of

rudeness and we rudeness and we

then measure 10 then measure 10

profs rudeness profs rudeness

index; once when index; once when

the offending TA the offending TA

is male, and once is male, and once

when they are when they are

female.female.

Female TA Male TA Difference Prof 1 15 10 5 Prof 2 22 20 2 Prof 3 18 19 -1 Prof 4 5 4 1 Prof 5 40 33 7 Prof 6 20 20 0 Prof 7 14 16 -2 Prof 8 10 14 -4 Prof 9 22 10 12 Prof 10 18 13 5 Mean 18.4 15.9 2.5 SD 9.29 7.88 4.79


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Question becomes, is the average Question becomes, is the average difference score significantly different difference score significantly different from 0?from 0?

So, when we do the math:So, when we do the math:

65.1

10

79.405.2

n

s0D

tD


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The critical t with alpha equal .05 The critical t with alpha equal .05 (two-tailed) and 9 (n-1) degrees of (two-tailed) and 9 (n-1) degrees of freedom is 2.262.freedom is 2.262.

Since tSince tobtobt is not greater than t is not greater than tcritcrit, we , we can not reject Hcan not reject H0.0.

Thus, we have no evidence that the Thus, we have no evidence that the profs rudeness is difference across profs rudeness is difference across TAs of different gendersTAs of different genders


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Pairs of Means –Pairs of Means –Independent SamplesIndependent Samples

Another common situation is one where Another common situation is one where we have two of more groups composed of we have two of more groups composed of independent observations.independent observations.

That is, each subject is in only one group That is, each subject is in only one group and there is no reason to believe that and there is no reason to believe that knowing about one subjects performance knowing about one subjects performance in one of the groups would tell you in one of the groups would tell you anything about another subjects anything about another subjects performance in one of the other groupsperformance in one of the other groups


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In this situation we are said to have In this situation we are said to have independent samples or, as it is sometimes independent samples or, as it is sometimes called, a between subjects designcalled, a between subjects design

ExampleExample: Let’s take the famous : Let’s take the famous “Misinformation Effect” memory experiment “Misinformation Effect” memory experiment where subjects see a video of a car accident where subjects see a video of a car accident and are asked to estimate the speed of the and are asked to estimate the speed of the car involved in the accident. The adjective car involved in the accident. The adjective used to describe the collision (smashed vs. used to describe the collision (smashed vs. ran into vs. contacted) is varied across ran into vs. contacted) is varied across groups (n=20). Did the manipulation affect groups (n=20). Did the manipulation affect speed estimates? That is, are the mean speed estimates? That is, are the mean speed estimates of the various groups speed estimates of the various groups different?different?


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In the accident video, about how fast (in In the accident video, about how fast (in km/h) do you think the gray car was km/h) do you think the gray car was going when it ________ the side of the red going when it ________ the side of the red car?car?

Group Mean SD 1) smashed into 74.0 9.06 2) ran into 67.2 10.55 3) made contact with 66.3 11.22


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There are, in fact, three different t-There are, in fact, three different t-tests we can perform in this situation, tests we can perform in this situation, comparing groups 1 &2, 1&3, or 2&3.comparing groups 1 &2, 1&3, or 2&3.

For demonstration purposes, let’s For demonstration purposes, let’s only worry about groups 1 & 2 for only worry about groups 1 & 2 for now.now.

So, we could ask, do subjects in Group So, we could ask, do subjects in Group 1 give different estimates of the gray 1 give different estimates of the gray car’s speed than subjects in Group 2?car’s speed than subjects in Group 2?


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Variance Sum LawVariance Sum Law

When testing a difference between When testing a difference between two independent means, we must two independent means, we must once again think about the once again think about the sampling distribution associated sampling distribution associated with Hwith H0.0.

If we assume the means come from If we assume the means come from separate populations, we could separate populations, we could simultaneously draw samples from simultaneously draw samples from each population and calculate the each population and calculate the mean of each samplemean of each sample


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If we repeat this process a number of If we repeat this process a number of times, we could generate sampling times, we could generate sampling distributions of the mean of each distributions of the mean of each population, and a sampling distribution population, and a sampling distribution of the difference of the two means.of the difference of the two means.

If we actually did this, we would find that If we actually did this, we would find that the sampling distribution of the the sampling distribution of the difference would have a variance equal to difference would have a variance equal to the sum of the two population variancesthe sum of the two population variances


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In fact, the Variance Sum Law states In fact, the Variance Sum Law states that:that:

The variance of a sum or difference of two The variance of a sum or difference of two independentindependent variables is equal to the sum variables is equal to the sum of their variancesof their variances

2

2

2

1xx2

21


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Now recall that when we performed a t-test Now recall that when we performed a t-test in the situation where the population in the situation where the population standard deviation was unknown, we used standard deviation was unknown, we used the formula:the formula:

Given all of the above, we can now alter this Given all of the above, we can now alter this formula in a way that will allow us to use it formula in a way that will allow us to use it in the independent means examplein the independent means example

n

sx

t


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Specifically, instead of comparing a Specifically, instead of comparing a single sample mean with some mean, we single sample mean with some mean, we want to see if the difference between two want to see if the difference between two sample means equals zero.sample means equals zero.

Thus the numerator (top part) will Thus the numerator (top part) will change to:change to:

or simplyor simply)xx(

0)xx(

21

21


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And, because the standard error And, because the standard error associated with the difference between associated with the difference between two means is the sum of each mean’s two means is the sum of each mean’s standard error (by the variance sum law), standard error (by the variance sum law), the denominator of the formula changes tothe denominator of the formula changes to

2

2

2

1

2

1

n

s

n

s


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Thus, the basic formula for calculating a Thus, the basic formula for calculating a t-test for independent samples is:t-test for independent samples is:

2

2

2

1

2

1

21

ns

ns

)xx(t


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Finishing the example:Finishing the example:

Comparing groups 1 and 2, we end up Comparing groups 1 and 2, we end up with:with:

df = (ndf = (n11+n+n22-2) = 38, t-2) = 38, tCRITCRIT = 2.021. = 2.021.

Since tSince tOBTOBT > t > tCRITCRIT we reject H we reject H00

19.211.3

8.6

2030.111

2008.82

2.670.74

ns

ns

)xx(t

2

2

2

1

2

1

21


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Pooling Variances & Unequal Pooling Variances & Unequal NsNs

The previous formula is fine when The previous formula is fine when sample sizes are equal.sample sizes are equal.

However, when sample sizes are However, when sample sizes are unequal, it treats both of the Sunequal, it treats both of the S22 as as equal in terms of their ability to equal in terms of their ability to estimate the population varianceestimate the population variance


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Instead, it would be better to combine Instead, it would be better to combine the s2 in a way that weighted them the s2 in a way that weighted them according to their respective sample according to their respective sample sizes. This is done using the following sizes. This is done using the following pooled variance estimate:pooled variance estimate:

2nn

s)1n(s)1n(s

21

2

22

2

112

p


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Given this, the new formula for Given this, the new formula for calculating an independent groups t-test calculating an independent groups t-test is:is:

)n1

n1

(s

)xx(t

21

2

p

21


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Using the pooled variances version Using the pooled variances version of the t formula for independent of the t formula for independent samples is no different from using samples is no different from using the separate variances version the separate variances version when sample sizes are equal. It when sample sizes are equal. It can have a big effect, however, can have a big effect, however, when sample sizes are unequal.when sample sizes are unequal.


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Heterogeneity of VarianceHeterogeneity of Variance

The text book has a large section on The text book has a large section on heterogeneity of variance (pp 213-216) heterogeneity of variance (pp 213-216) including lots of nasty looking formulae. including lots of nasty looking formulae. All I want you to know is the following:All I want you to know is the following:

When doing a t-test across two groups, When doing a t-test across two groups, you are assuming that the variances of you are assuming that the variances of the two groups are approximately equal.the two groups are approximately equal.

If the variances look fairly different, If the variances look fairly different, there are tests that can be used to see if there are tests that can be used to see if the difference is so great as to be a the difference is so great as to be a problemproblem


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Heterogeneity of VarianceHeterogeneity of Variance

If the variances are different across If the variances are different across the groups, there are ways of the groups, there are ways of correcting the t-test to take the correcting the t-test to take the heterogeneity in account.heterogeneity in account.

In fact, t-tests are often quite In fact, t-tests are often quite robust to this problem, so you robust to this problem, so you don’t have to worry about it too don’t have to worry about it too much.much.


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The CookbookThe Cookbook

One Observation vs. Population One Observation vs. Population ::

x

z


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One Mean vs. One Population MeanOne Mean vs. One Population Mean::

Population variance known:Population variance known:

n

xz


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One Mean vs. One Population MeanOne Mean vs. One Population Mean::

Population variance unknown:Population variance unknown:

df = n-1df = n-1

n

sx

t


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Two MeansTwo Means::

Matched samples:Matched samples:

first create a difference score, then...first create a difference score, then...

df = ndf = nDD-1-1

n

s0D

tD


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Independent samples:Independent samples:

df = ndf = n11+n+n22-2-2

2

2

2

1

2

1

21

ns

ns

)xx(t


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Independent samples. . .continued:Independent samples. . .continued:

where:where:

Easy as baking a cake, right? Now for Easy as baking a cake, right? Now for some examples of using these recipes to some examples of using these recipes to cook up some tasty conclusions. . .cook up some tasty conclusions. . .

2nn

s)1n(s)1n(s

21

2

22

2

112

p


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ExamplesExamples

1) The population spends an average of 8 hours per 1) The population spends an average of 8 hours per day working, with a standard deviation of 1 hour. day working, with a standard deviation of 1 hour. A certain researcher believes that profs work less A certain researcher believes that profs work less hours than average and wants to test whether hours than average and wants to test whether the average hours per day that profs work is the average hours per day that profs work is different from the population. This researcher different from the population. This researcher samples 10 professors and asks them how many samples 10 professors and asks them how many hours they work per day, leading to the following hours they work per day, leading to the following data set:data set:

perform the appropriate statistical test and state perform the appropriate statistical test and state your conclusions.your conclusions.

6, 12, 8, 15, 9, 16, 7, 6, 14, 15


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ExamplesExamples

2) 2) Now answer the question again except Now answer the question again except assume the population variance is unknown.assume the population variance is unknown.

3) Does the use of examples improve memory 3) Does the use of examples improve memory for the concepts being taught? Joe for the concepts being taught? Joe Researcher tested this possibility by Researcher tested this possibility by teaching 10 subjects 20 concepts each. For teaching 10 subjects 20 concepts each. For each subject, examples were provided to each subject, examples were provided to help explain 10 of the new concepts, no help explain 10 of the new concepts, no examples were provided for the other 10. examples were provided for the other 10. Joe then tested his subjects memory for the Joe then tested his subjects memory for the concepts and recorded how many concepts, concepts and recorded how many concepts, out of 10, that the subject could remember. out of 10, that the subject could remember. Here is the data (next slide):Here is the data (next slide):


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ExamplesExamples

Subject No Example Example

1 6 8 2 8 8 3 5 6 4 4 6 5 7 7 6 8 7 7 2 5 8 5 7 9 6 7 10 8 9


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ExamplesExamples

4) Circadian rhythms suggest that 4) Circadian rhythms suggest that young adults are at their young adults are at their physical peek in the early physical peek in the early afternoon, and are at their afternoon, and are at their physical low point in the early physical low point in the early morning. Are cognitive factors morning. Are cognitive factors affected by these rhythms? To affected by these rhythms? To test this question I bring test this question I bring subjects in to run a recognition subjects in to run a recognition memory experiment. Half of the memory experiment. Half of the subjects are run at 8 am, the subjects are run at 8 am, the other other half at 2pm. I then record their half at 2pm. I then record their recognition memory accuracy. recognition memory accuracy. Here are the results:Here are the results:

8 am 2 8 am 2 pmpm

.60.60 .78 .78 .58.58 .85 .85 .68.68 .81 .81 .74.74 .82 .82 .71.71 .76 .76 .62.62 .73 .73

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Psy B07 Chapter 7Slide 1 HYPOTHESIS TESTING APPLIED TO MEANS