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Week 2Week 2

Sampling distributions and Sampling distributions and testing hypotheses testing hypotheses

handout available at handout available at http://homepages.gold.ac.uk/aphomehttp://homepages.gold.ac.uk/aphome

8-10-2007Trevor ThompsonTrevor Thompson

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Review of following topics:Review of following topics: 1) How individual scores are distributed1) How individual scores are distributed

2) How 2) How meanmean scores are distributed scores are distributed

3) One-sample z-test3) One-sample z-test testing the difference between a sample mean and a known population meantesting the difference between a sample mean and a known population mean

- Howell (2002) Chap 4 & 7. ‘Statistical Methods for - Howell (2002) Chap 4 & 7. ‘Statistical Methods for Psychology’Psychology’

1) How individual scores are distributed1) How individual scores are distributed

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Distribution of individual Distribution of individual scoresscores How individual values are distributed depends How individual values are distributed depends

on the nature of these valueson the nature of these values Uniform distributionUniform distribution

Dice scores are uniformlyDice scores are uniformlydistributed (each score has distributed (each score has an equal probability of an equal probability of occurrence)occurrence)

Normal distributionNormal distributionScores on many variablesScores on many variablesare normallyare normallydistributed (e.g. IQ)distributed (e.g. IQ)

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Sampling distribution is not always identical to Sampling distribution is not always identical to population distribution because of ‘sampling error’ population distribution because of ‘sampling error’

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Distribution of individual Distribution of individual scoresscores

What is probability (p-value) of What is probability (p-value) of randomly sampling one person randomly sampling one person with an IQ of 100 or more?with an IQ of 100 or more? ≈ ≈ 50%50%

What is probability of randomly selecting an IQ What is probability of randomly selecting an IQ score of 130+? (We can calculate this from what score of 130+? (We can calculate this from what we know about the properties of the normal we know about the properties of the normal curve)curve)

The probability for any IQ can be calculated* – The probability for any IQ can be calculated* – calculate the z-score (i.e. the number of SD’s above calculate the z-score (i.e. the number of SD’s above or below the mean), then look up corresponding p-or below the mean), then look up corresponding p-value in table (or use SPSS CDF function)value in table (or use SPSS CDF function)

(*assuming population parameters of M=100, SD=15 and normal distribution)(*assuming population parameters of M=100, SD=15 and normal distribution)

≈ ≈ 2.5%2.5%

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1) How individual scores are distributed1) How individual scores are distributed

2) How 2) How meanmean scores are distributed scores are distributed

3) One-sample z-test3) One-sample z-test

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Sampling distribution of Sampling distribution of meansmeans

Q: What is the probability of a group of 36 having a mean IQ of 106 ?

We need to know how means are We need to know how means are distributed to answer this questiondistributed to answer this question

Specifically, we need to know (as Specifically, we need to know (as with individual scores):with individual scores): 1) Mean1) Mean 2) Shape of distribution2) Shape of distribution 3) Standard deviation3) Standard deviation

same as mean of individual scores (100) next slide

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Shape of distribution of Shape of distribution of meansmeansI repeatedly sampled 36 scores and calculated the mean. I then

repeated this several thousand times and plotted these means:

Sample 1: Random sample of 36 scores produced M=103.5Sample 2: Random sample of 36 scores produced M=100Sample 3: Random sample of 36 scores produced M=96Sample 4: Random sample of 36 scores produced M=102Sample 5: Random sample of 36 scores produced M=100

The distribution of means appears to be the same as individual scores!– i.e. normal

95 97.5 100 102.5 105

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1) Mean – equal to the population mean (100)1) Mean – equal to the population mean (100) 2) Shape - normal2) Shape - normal 3) Standard deviation – how widely are the 3) Standard deviation – how widely are the

means spread?means spread?

Sampling distribution of Sampling distribution of meansmeans

Mean scores are spread more closely Mean scores are spread more closely around the centre than individual scoresaround the centre than individual scores this makes intuitive sense – while individual this makes intuitive sense – while individual

scores of 130 are not exceptionally rare scores of 130 are not exceptionally rare (p=2.5%), mean IQs of 130 would be (p=2.5%), mean IQs of 130 would be extremely rare when group size is 1,000!extremely rare when group size is 1,000!

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Sampling distribution of Sampling distribution of meansmeans

In fact, we can calculate precisely the spread of In fact, we can calculate precisely the spread of mean mean scoresscores around the centre: around the centre:

SEM= SxSEM= Sx √N √N

SEM (the SEM (the sstandard tandard eerror of the rror of the mmean) ean) is the standard is the standard deviation of the deviation of the meanmean (rather than standard deviation (rather than standard deviation ofof individual scores)individual scores)

The above formula shows that the bigger the sample size The above formula shows that the bigger the sample size (N), the smaller the SEM – i.e. the more closely scores are (N), the smaller the SEM – i.e. the more closely scores are clustered around the population meanclustered around the population mean

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Sampling distribution of Sampling distribution of meansmeans

Q: What is the probability of N=36 having a mean IQ of at least 106 ? We can now plot the sampling distribution of the We can now plot the sampling distribution of the meansmeans. We know the shape is normal, M=100 &. We know the shape is normal, M=100 &

SEMSEM =1515 = 2.52.5

√√3636

So, if we know how individual scores are distributed (i.e. shape, M & SD) we also know how means are distributed and can test hypotheses about groups

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Confidence LimitsConfidence Limits The 95% confidence limits is the range within The 95% confidence limits is the range within

which 95% of the sample means will fallwhich 95% of the sample means will fall If a sample mean lies within these limits then If a sample mean lies within these limits then

we cannot reject the null hypothesis we cannot reject the null hypothesis

Our value of 106 lies outside these limits –we Our value of 106 lies outside these limits –we reject the null hypothesis (p<.05!)reject the null hypothesis (p<.05!)

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1) How individual scores are distributed1) How individual scores are distributed

2) How 2) How meanmean scores are distributed scores are distributed

3) One-sample z-test3) One-sample z-test

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One sample z-testOne sample z-test One sample z-test: Compares the mean score of One sample z-test: Compares the mean score of

one group against a population mean. This can one group against a population mean. This can only be performed when we know the population only be performed when we know the population mean and the population standard deviationmean and the population standard deviation

To perform a one-sample z-testTo perform a one-sample z-test, calculate how , calculate how many SEMs above/below the population mean many SEMs above/below the population mean your sample mean is. Expressed as a formula: your sample mean is. Expressed as a formula: z= X – μ (where SEM=z= X – μ (where SEM=σσ/√N)/√N)

SEMSEM

A one-sample z-test is what we have previously A one-sample z-test is what we have previously performed! performed! z= (106 – 100)/2.5 z= (106 – 100)/2.5

z=2.4, which gives p<.05 –significant! z=2.4, which gives p<.05 –significant!

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One sample t-testOne sample t-test A one sample z-test is used when we already

know the population mean and SD

A one sample t-test is used when we know the population mean (μ) but not the population SD (σ)

As we do not know σ, we estimate it from s (sample SD). But, as s is often too small, the p-value is inaccurate when using z-distribution tables. Use t-distribution tables for more conservative p-values

To perform a one-sample t-test: t = X – μt = X – μ (where SEM=s/√N)(where SEM=s/√N) SEMSEM

but look up p-value from a t not a z-distribution table

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Central Limit TheoremCentral Limit Theorem Everything we have done so far is explained by central limit theoremEverything we have done so far is explained by central limit theorem

‘‘Given a population with mean, Given a population with mean, , and standard deviation, , and standard deviation, , the sampling distribution of the mean will have’:, the sampling distribution of the mean will have’:

(i) a mean equal to (i) a mean equal to

(ii) a standard deviation equal to (ii) a standard deviation equal to /√N, where N is /√N, where N is the sample sizethe sample size

(iii) a distribution which will approach the normal (iii) a distribution which will approach the normal distribution as N increasesdistribution as N increases

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Central Limit TheoremCentral Limit Theorem The approximation of the sampling distribution of The approximation of the sampling distribution of

the mean to a normal distribution is true -whatever the mean to a normal distribution is true -whatever the shape of the distribution of individual valuesthe shape of the distribution of individual values

Population

Samp. Dist. of Mean

Normal Normal for all N

Symmetric and unimodal Normal for s mall N Highly skewed Normal for N>30

distribution of single die scores

distribution of mean dice scores

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One sample z-tests - One sample z-tests - examplesexamples

1) The mean IQ of a group of 16 people was measured as 103. Is this 1) The mean IQ of a group of 16 people was measured as 103. Is this significantly different from the population using the population significantly different from the population using the population parameters previous specified?parameters previous specified?

NoNo, SEM=3.75 (15/√16), SEM=3.75 (15/√16)

z=0.8 (103-100/3.75)z=0.8 (103-100/3.75)

p>0.5 – non-significantp>0.5 – non-significant

2) Is a sample of 25 dice throws, with a mean score 2) Is a sample of 25 dice throws, with a mean score of 3.85, sampled from a fair die? [of 3.85, sampled from a fair die? [=3.5, =3.5, =1.7]=1.7]

No,No, SEM=0.34 (1.7/√25) SEM=0.34 (1.7/√25)

zz≈≈1 (3.85-3.5/0.34)1 (3.85-3.5/0.34)

p>0.5 – non-significantp>0.5 – non-significant

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SummarySummary How to perform a one sample z-testHow to perform a one sample z-test

How to perform a one sample t-testHow to perform a one sample t-test

Underlying logic behind one sample testsUnderlying logic behind one sample tests

Rules of central limit theoremRules of central limit theorem