Testing Hypotheses About a Mean1

8/3/2019 Testing Hypotheses About a Mean1

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Testing Hypotheses about a Mean


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Testing a Hypothesis about a Mean

Given a population ( , ), we saw that when weobtain a sample of n scores through randomsampling and computed the sample mean of ourscores, the means would vary from sample to sample.

How do we infer the value of the population mean( ) when our sample mean is only one of many

different sample means?Translate research question into a statisticalhypothesis.


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Large, national population of 6th grade students thattook a math achievement test. Test manual indicatesthat the national norm for 6th grade students is 85.

From our district, we have a sample ( n = 100) of 6thgrade students that complete the same test. Are theyat the same level as students in the population?

Statistical hypothesis: H 0 : X = 85.The population mean of the math achievement scores inour district is 85.


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Hypotheses are always about parameters NOTstatistics.

H0 is for null hypothesis. It is tested by the

researcher.We can reject or fail to reject (retain) H 0 .

Null hypothesis is NOT synonymous with zero or nilhypothesis.

Every H 0 has an alternative hypothesis (H A). It is thehypothesis that the researcher hopes is true.

HA: X 85.

Examples of alternative hypotheses (H A).


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When do we reject or fail to reject H 0?If a sample mean is so different from what is expectedwhen H 0 is true that it would be so unlikely to haveoccurred by chance, reject H 0 . Otherwise, do not reject H 0 .

What is the criterion for unlikely or unusual?The criterion is called the level of significance symbolized

by alpha ( ). Often, .05.


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General Steps for Testing Hypotheses

1. Specify H 0 and H A about a parameter of the population.

2. Random sample drawn from population and the value ofthe sample statistic is calculated.

3. Examine random sampling distribution of the samplestatistic to learn what sample outcomes would occur bychance over an infinite number of repetitions (and with

what probability) if H 0 is true.4. H 0 is retained if the sample statistic is in line with the

outcomes expected if H 0 is true. Otherwise, reject H 0 andaccept H A.


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Returning to our example:

H0 : X = 85

HA: X 85

Sample of 6th grade students ( n = 100). The sample meanon the math achievement test for these students was 90.

We will adopt the conventional .05 level of significance.

We will reject H 0 only if our sample mean is so deviant that itfalls in the upper .025 or lower .025 of all possible samplemeans that would occur by chance when H 0 is true.

Fail to reject H 0 if our sample mean falls in the central .95.


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Critical values

Critical values: value(s) that separates the region ofrejection from the region of retention. In our case,look under Table A in Appendix D for the critical z

values.Region of rejection: portion of the samplingdistribution that leads to rejection of H 0 .

Region of retention: portion of the samplingdistribution that leads to retention of H 0 .


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We have a sample mean of 90. But, we still need todetermine the relative position of our sample mean.

How do we do this?

Assume = 20.

X X

X

X X z

n


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Assume = 20.

90 852.520

100

X X

X

X X z

n


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Conclusion: Reject H 0 . It is unreasonable to believe that themean of the population from which the sample came is 85.


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Rejecting versus Retaining H 0

Rejecting H 0 indicates that it does not seemreasonable to believe that H 0 is true.

Failing to reject (or retaining) H 0 indicates that we

do not have sufficient evidence to reject H 0 .NEVER say that we accept H 0 .


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Nondirectional hypothesis test. Two-tailed test.Possible to detect difference between true value andhypothesized value of the parameter regardless of

direction of the difference.See previous example

Directional hypothesis test. One-tailed test.Interest lies in detecting differences in a particulardirection. For example:

H0 : X = 100

HA: X < 100


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For our one-tailed test, if our sample mean was 130,would we reject H 0 or fail to reject H 0? Why?


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In the example involving 6th grade students, wecomputed a z statistic associated with thecorresponding sample mean. Here, was known.

If is unknown, we use a different statistic called a t statistic.

Because is unknown, we must estimate with s (the

unbiased sample standard deviation).Thus, instead of converting our sample mean to a z ,we are converting our sample mean to a t .


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When is Unknown

X X

X

X X z

n

X X

X

X X t ss

n

Instead of:

We use:


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When is Unknown

X X

X

X X z

n

X X

X

X X t ss

n

Because is unknown, theStandard Error of theMean is also unknown.

By using s, we have anEstimate of the StandardError of the Mean.


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t distribution

The t statistic is an approximation of z .

t is not normally distributed.

The proper distribution of t was discovered by WilliamS. Gosset who published under the pseudonym Student.

Gosset discovered this distribution while working atGuinness Brewing Company. The distribution came to beknown as Students distribution of t .


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Characteristics of the t Distribution

Not a single distribution, but a family of distributions. Eachdiffers in its approximation to the standard normal curve.

Exact shape of a t distribution depends on the sample size,or more specifically, the degrees of freedom ( df ). Whentesting a hypothesis about a single mean, df = n 1.

Because Students distribution of t is an approximation of

z , not surprisingly, t shares some similarities to z .Mean of 0.

Unimodal.

Symmetric.


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Characteristics of the t Distribution

Differences between t and z . The t distribution:is narrower at the peak and it has fatter tails than z .

has a larger standard deviation than z .What is the standard deviation of the z distribution?

depends on df .Consider a hypothesis test about a mean when n = 5. df =?

Consider a hypothesis test about a mean when n = 13. df =?

Consider when we have an infinite sample size. df = .

As N increases, the variance of the t distribution approachesthe variance of the standard normal distribution 1


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t Distribution

In summary, when is unknown, we will use t to testa hypothesis about a mean.

Instead of referring to the z table, we will refer to

the table for Students distribution of t (Table D inAppendix D). To use this table, need to know:

df

whether our test is one-tailed or two-tailed


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Example 1

Background: Based on a babys chronologic age,there are various psychomotor skills that a babyshould attain (e.g., looking at own hand, able tograsp objects voluntarily, roll from front to back,etc.). On the index below, we will assume thathigher scores suggest more impairment while lower

scores suggest having advanced psychomotordevelopment (relative to age).


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Example 1

Sample of 56 low birth weight babies. ThePsychomotor Development Index (PDI) wasadministered when 6 months old. For a normal

population of babies, the mean on the PDI is 100.For our sample of babies, the mean PDI score was104.125. The sample standard deviation of the PDI

scores was 12.584.Does the population mean from which our samplecame differ from that of a normal population?

Assume = .05.


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Example 1

H0 : = 100

HA: 100

Table D in Appendix D. df = 55. Critical t = 2.004.

104.125 1002.45

12.58456

X X

X

X X t

ssn


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Example 1

Conclusion: Reject H 0 . The population from which oursample was drawn differs significantly from thenormal population of children on the PDI.

It appears that the population from which oursample was drawn is more impaired in terms oftheir psychomotor skills.


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Example 2

Background: A Health Director at a local universitybelieves that students at her campus are very health-conscious and, as a result, tend to consume less sugar

than do most people living in the U.S. The directorknows that the average person in the U.S. consumes 100lbs. of sugar a year (mostly in the form of soft drinks,candy, pastries).


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Example 2

The director decides to obtain a random sample of 25students enrolled at the university and determine thequantity of sugar consumed by each student during a

2-week period. (Note that yearly sugar consumptioncan be calculated by multiplying the 2-week total by26.) Assume = .05.

The mean yearly amount of sugar consumed by the

25 students is 80 (= 3.0769 x 26).The sample standard deviation for the yearly amountof sugar consumed by the 25 students is 35.5.


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Example 2

H0 : = 100

HA: < 100

Table D in Appendix D. df = 24. Critical t = 1.711.

80 1002.8169

35.525

X X

X

X X t

ssn


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Example 2

Conclusion: Reject H 0 . The population from which oursample was drawn differs significantly from thepopulation mean yearly amount of sugar consumed

in the U.S.The population of students at the university are morehealth conscious than the typical person in the U.S.

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Testing Hypotheses About a Mean1