© asup-2007 inference about means and mean differences 1 chapter 9 introduction to t statistic

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Inference about Means and Mean Differences

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Chapter 9INTRODUCTION TO t STATISTIC

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Preview-1 In the previous chapter, we presented

the statistical procedure that permit researcher to use sample mean to test hypothesis about an unknown population

Remember that the expected value of the distribution of sample means is μ, the population mean

σM = σ√n

z =M - μ

σM

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THE PROBLEM WITH z-SCORE A z-score requires that we know the

value of the population standard deviation (or variance), which is needed to compute the standard error

In most situation, however, the standard deviation for the population is not known

In this case, we cannot compute the standard error and z-score for hypothesis test. We use t statistic for hypothesis testing when the population standard deviation is unknown

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THE t STATISTIC:AN ALTERNATIVE TO z

The goal of the hypothesis test is to determine whether or not the obtained result is significantly

greater than would be expected by chance.

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Introducing t Statistic

σM =σ√n

Now we will estimates the standard error by simply substituting the sample variance or standard deviation in place of the unknown population value

SM =s√n

Notice that the symbol for estimated standard error of M is SM instead of

σM , indicating that the estimated value is computed from sample data rather than from the actual population parameter

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z-score and t statistic

σM = σ√n

z =M - μ

σM

SM = s√n

t =M - μ

SM

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The t Distribution Every sample from a population can be

used to compute a z-score or a statistic If you select all possible samples of a

particular size (n), then the entire set of resulting z-scores will form a z-score distribution

In the same way, the set of all possible t statistic will form a t distribution

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The Shape of the t Distribution The exact shape of a t distribution

changes with degree of freedom There is a different sampling

distribution of t (a distribution of all possible sample t values) for each possible number of degrees of freedom

As df gets very large, then t distribution gets closer in shape to a normal z-score distribution

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HYPOTHESIS TESTS WITH t STATISTIC

The goal is to use a sample from the treated population (a treated sample) as the determining whether or not the treatment has any effectKnown population before treatment

Unknown population after treatment

μ = 30 μ = ?

TREATMENT

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HYPOTHESIS TESTS WITH t STATISTIC As always, the null hypothesis states that the

treatment has no effect; specifically H0 states that the population mean is unchanged

The sample data provides a specific value for the sample mean; the variance and estimated standard error are computed

t =sample mean

(from data)

Estimated standard error (computed from the sample data)

population mean (hypothesized from H0)-

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A psychologist has prepared an “Optimism Test” that is administered yearly to graduating college seniors. The test measures how each graduating class feels about it future. The higher the score, the more optimistic the class. Last year’s class had a mean score of μ = 19. A sample of n = 9 seniors from this years class was selected and tested. The scores for these seniors are as follow:

19 24 23 27 19 20 27 21 18On the basis of this sample, can the psychologist

conclude that this year’s class has a different level of optimism than last year’s class?

LEARNING CHECK

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STEP-1: State the Hypothesis, and select an alpha level

H0 : μ = 19 (there is no change)

H1 : μ ≠ 19 (this year’s mean is different)

Example we use α = .05 two tail

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STEP-2: Locate the critical region Remember that for hypothesis test with t

statistic, we must consult the t distribution table to find the critical t value. With a sample of n = 9 students, the t statistic will have degrees of freedom equal to

df = n – 1 = 9 – 1 = 8 For a two tailed test with α = .05 and df =

8, the critical values are t = ± 2.306. The obtained t value must be more extreme than either of these critical values to reject H0

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STEP-3: Obtain the sample data, and compute the test statistic

Find the sample mean

Find the sample variances

Find the estimated standard error SM

Find the t statistic

SM = s√n

t =M - μ

SM

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STEP-4: Make a decision about H0, and state conclusion

The obtained t statistic (t = 2.626) is in the critical region. Thus our sample data are unusual enough to reject the null hypothesis at the .05 level of significance.

We can conclude that there is a significant difference in level of optimism between this year’s and last year’s graduating classes

t(8) = 2.626, p<.05, two tailed

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The critical region in thet distribution for α = .05 and df

= 8

Reject H0 Reject H0

Fail to reject H0

-2.306 2.306

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DIRECTIONAL HYPOTHESES AND ONE-TAILED TEST

The non directional (two-tailed) test is more commonly used than the directional (one-tailed) alternative

On other hand, a directional test may be used in some research situations, such as exploratory investigation or pilot studies or when there is a priori justification (for example, a theory previous findings)

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A fund raiser for a charitable organization has set a goal of averaging at least $ 25 per donation. To see if the goal is being met, a random sample of

recent donation is selected.The data for this sample are as follows:20 50 30 25 15 20 40 50 10 20

LEARNING CHECK

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= 9

Reject H0

Fail to reject H0

1.883

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Chapter 10THE t TEST FOR TWO

INDEPENDENT SAMPLES

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Preview-2 In many research situations, however,

its difficult or impossible for a researcher to satisfy completely the rigorous requirement of an experiment

In these situations, a researcher can often devise a research strategy (a method of collecting data) that is similar to an experiment but fails to satisfy at least one of the requirement of a true experiment

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Inference about Means and Mean Differences NonExperimental and Quasi

Experimental Although these studies resemble

experiment, they always contain a confounding variable or other threat to internal validity that is an integral part of the design and simply cannot be removed

The existence of a confounding variable means that these studies cannot establish unambiguous cause-and-effect relationship and, therefore, are not true experiment

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Experimental … is the degree to which the research

strategy limits the confounding and control threats to internal validity

If a research design makes little or no attempt to minimize threats, it is classified as nonexperimental

A quasi experimental design makes some attempt to minimize threats to internal validity and approach the rigor of a true experiment

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In an experiment… … a researcher typically creates

treatment condition by manipulating an IV, then measures participants to obtain a set of scores within each condition

If the score in one condition are significantly different from the other score in another condition, the researcher can conclude that the two treatment condition have different effects

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Experimental Similarly, a nonexperimental study

also produces group of scores to be compared for significant differences

One variable is used to create groups or conditions, then a second variable is measured to obtain a set of scores within each condition

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Experimental In nonexperimental and quasi-

experimental studies, the different groups or conditions are not created by manipulating an IV

The groups usually defined in terms of a preexisting participant variable (male/female) or in term of time (before/after)

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Single sample techniques are used occasionally in real research, most research studies require the comparison of two (or more) sets of data

There are two general research strategies that can be used to obtain of the two sets of data to be compared:○ The two sets of data come from the two

completely separate samples (independent-measures or between-subjects design)

○ The two sets of data could both come from the same sample (repeated-measures or within subject design)

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Do the achievement scores for students taught by method A differ from the scores for students taught by method B?In statistical terms, are the two population means the same or different?

Unknownµ =?

SampleA

Unknownµ =?

SampleB

Taught by

Method A

Taught by

Method B

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THE HYPOTHESES FOR AN INDEPENDENT-MEASURES TEST

The goal of an independent-measures research study is to evaluate the mean difference between two population (or between two treatment conditions)

H0: µ1 - µ2 = 0 (No difference between the population means)

H1: µ1 - µ2 ≠ 0 (There is a mean difference)

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THE FORMULA FOR AN INDEPENDENT-MEASURES

HYPOTHESIS TEST

In this formula, the value of M1 – M2 is obtained from the sample data and the value for µ1 - µ2 comes from the null hypothesis

The null hypothesis sets the population mean different equal to zero, so the independent-measures t formula can be simplifier further

t =sample mean

difference

estimated standard error

population mean difference-

=M1 – M2

S (M1 – M2)

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THE STANDARD ERROR

To develop the formula for S(M1 – M2) we will consider the following points:

Each of the two sample means represent its own population mean, but in each case there is some error

SM = s2

n√SM1-M2 = s1

2

n1√s2

2

n2+

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POOLED VARIANCE The standard error is limited to

situation in which the two samples are exactly the same size (that is n1 – n2)

In situations in which the two sample size are different, the formula is biased and, therefore, inappropriate

The bias come from the fact that the formula treats the two sample variance

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POOLED VARIANCE for the independent-measure t

statistic, there are two SS values and two df values

SP2 = SS

nSM1-M2 = s1

2

n1√s2

2

n2+

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HYPOTHESIS TEST WITH THE INDEPENDENT-MEASURES t

STATISTICIn a study of jury behavior, two samples of participants were provided details about a trial in which the defendant was obviously

guilty. Although Group-2 received the same details as Group-1, the second group was also

told that some evidence had been withheld from the jury by the judge. Later participants were asked to recommend a jail sentence. The length of term suggested by each participant is presented. Is there a significant difference between the two groups in their responses?

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THE LENGTH OF TERM SUGGESTED BY EACH

PARTICIPANTGroup-1 scores: 4 4 3 2 5 1 1 4Group-2 scores: 3 7 8 5 4 7 6 8

There are two separate samples in this study. Therefore the analysis will use the independent-measure t test

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STEP-1: State the Hypothesis, and select an alpha level

H0 : μ1 - μ2 = 0 (for the population, knowing evidence has been withheld has no effect on the suggested sentence)

H1 : μ1 - μ2 ≠ 0 (for the population, knowledge of withheld evidence has an effect on the jury’s response)

We will set α = .05 two tail

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STEP-2: Identify the critical region For the independent-measure t statistic,

degrees of freedom are determined bydf = n1 + n2 – 2 = 8 + 8 – 2 = 14

The t distribution table is consulted, for a two tailed test with α = .05 and df = 14, the critical values are t = ± 2.145.

The obtained t value must be more extreme than either of these critical values to reject H0

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STEP-3: Compute the test statistic

Find the sample mean for each groupM1 = 3 and M2 = 6

Find the SS for each groupSS1 = 16 and SS2 = 24

Find the pooled variance, andSP

2 = 2.86

Find estimated standard errorS(M1-M2) = 0.85

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STEP-4: Make a decision about H0, and state conclusion

The obtained t statistic (t = -3.53) is in the critical region on the left tail (critical t = ± 2.145). Therefore, the null hypothesis is rejected.

The participants that were informed about the withheld evidence gave significantly longer sentences,

t(14) = -3.55, p<.05, two tails

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LEARNING CHECK

The following data are from two separate independent-measures experiments. Without doing any calculation, which experiment is more likely to demonstrate a significant difference between treatment A and B? Explain your answer.

EXPERIMENT A EXPERIMENT BTreatment

ATreatment

BTreatment

ATreatment

B

n = 10 n = 10 n = 10 n = 10M = 42 M = 52 M = 61 M = 71

SS = 180 SS = 120 SS = 986 SS = 1042

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A psychologist studying human memory, would like to examine the process of

forgetting. One group of participants is required to memorize a list of words in the evening just before going to bed.

Their recall is tested 10 hours latter in the morning. Participants in the second group memorized the same list of words in he morning, and then their memories tested 10 hours later after being awake

all day.

LEARNING CHECK

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LEARNING CHECK

The psychologist hypothesizes that there will be less forgetting during less forgetting during sleep than a busy day. The recall scores for two samples of college students are follows:

Asleep Scores Awake Scores

15 13 14 14 15 13 14 12

16 15 16 15 14 13 11 12

16 15 17 14 13 13 12 14

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Sketch a frequency distribution for the ‘asleep’ group. On the same graph (in different color), sketch the distribution for the ‘awake’ group.Just by looking at these two distributions, would you predict a significant differences between two treatment conditions?

Use the independent-measures t statistic to determines whether there is a significant difference between the treatments. Conduct the test with α = .05

LEARNING CHECK

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OVERVIEW With a repeated-measures design, two

sets of data are obtained from the same sample of individuals

The main advantage of a repeated-measures design is that it uses exactly the same individual in all treatment conditions.

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Inference about Means and Mean Differences The Hypotheses for a Related-Samples

Test As always, the null hypotheses states that

for the general population there is no effect, no change, or no difference. H0: X2 - X1 = μD = 0

The alternative hypotheses states that there is a treatment effect that causes the scores in one treatment condition to be systematically higher (or lower) than the scores in the other condition. In symbols H1: μD ≠ 0

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Inference about Means and Mean Differences The t Statistic for Related

Samples The t statistic for related samples is

structurally similar to the other t statistics

One major distinction of the related samples t is that is based on difference scores rather than raw scores (X values)

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Inference about Means and Mean Differences The t Statistic for Related

Samples

or

t =sample statistic

estimated standard error

population parameter-

=MD – μD

SMD

S2 =SSn-1

=SSdf

S =SSdf√

SMD = S2

n-1√or SMD

=S

df√

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LEARNING CHECKPeople with agoraphobia are so filled with

anxiety about being in public places that they seldom leave their homes. Knowing this is a difficulty disorder to treat, a researcher tries a long-term treatment.A sample of individuals report how often they have ventured out of the house in the past month. Then they have receive relaxation training and are introduce to trips away from the house at gradually increasing durations.After 2 months of treatment, participants report the number of trip out of the house they made in the last 30 days.

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Person Before (X1) After (X2) Difference (D)

A

B

C

D

E

F

G

0

0

3

3

2

0

0

4

0

14

23

9

8

6

?

?

?

?

?

?

?Does the treatment have a significant effect on the number of trips a person takes?Test with α = .05 two tails

© asup-2007 inference about means and mean differences 1 chapter 9 introduction to t statistic

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