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  • 00:00

    DANIEL LITTLE: In this video, I willdiscuss the t-statistic and various testswhich utilize the t-statistic.The t-test is an inferential hypothesis testthat we use on a single sample of dataor on two samples of data to tell us some informationabout an unobserved population.

  • 00:23

    DANIEL LITTLE [continued]: With a single sample of data, we wantto know whether that sample was generatedfrom a specific hypothesized distribution.This hypothesis will typically justbe a mean value representing wherewe think the mean of our sample should lie.For instance, we may be interested in knowingwhether people are getting smarter over time.So our starting hypothesis would be

  • 00:44

    DANIEL LITTLE [continued]: that the mean of the newly tested sampleshould be equal to the current population mean.For something like IQ, this would be a hypothesisthat our IQ of our sample should equal100, which is the average IQ value basedon a long-term sample of IQ tests.

  • 01:08

    DANIEL LITTLE [continued]: If we know what the population standard deviation is, thenwe compute the probability of a sample mean by using a z-test.The z-statistic is computed by subtracting the populationmean, which is here represented by the Greek symbol, mu,from our observed sample mean and then dividing

  • 01:31

    DANIEL LITTLE [continued]: by the standard error of the mean.We use the standard error because we do not have accessto the entire population.However, we know from the central limit theoremthat the sample will have a standard deviation of sigma,which is this Greek symbol here, divided

  • 01:55

    DANIEL LITTLE [continued]: by the square root of n, which is the size of our sample.Sigma is our population standard deviation.As our sample size gets larger, the samplewill approach a normal distribution.So we can use the normal distributionto estimate the probability that the sample came

  • 02:16

    DANIEL LITTLE [continued]: from that particular hypothesized distribution.To find the probability of the z value usingthe normal distribution, we find the area under the curve.This represents the probability of a range of values.Consequently, we can tell where our observed z value

  • 02:38

    DANIEL LITTLE [continued]: lies in relation to all other possible z values.For instance, if our z value is greater than twostandard deviations above the mean,then the z value is approximately greater than 95percent of all possible z values.So 95% of our z values lie in the center portion

  • 02:60

    DANIEL LITTLE [continued]: of this normal distribution.As you may have anticipated, however,to compute the z-score, we need to know the population standarddeviation.Typically, we do not have this information.Consequently, we cannot compute the z-statistic.However, we can use a related statistic

  • 03:22

    DANIEL LITTLE [continued]: called the t-statistic by using the standard deviationof our sample in place of the standard deviationof the population.The computation of the t-statisticis very similar to the computation of z-statistic.We again, subtract the hypothesizedmean from the sample mean.So our hypothesized mean is, again,

  • 03:43

    DANIEL LITTLE [continued]: represented by the Greek letter, mu.Our sample mean is represented by the capital letter M,and we divide by the standard error of the sample mean.We compute the standard error of the samplemean by taking the square root of the sample standarddeviation, squaring it to get the variance,

  • 04:08

    DANIEL LITTLE [continued]: and then dividing by our sample size here represented by n.Once we arrive at the t-statistic,we also need to compute the degrees of freedom.Knowing the degrees of freedom willallow us to use the t-distributionto compute the probability of our t value.

  • 04:30

    DANIEL LITTLE [continued]: For a single sample t-test, our degrees of freedomis the size of our sample minus 1.n is the number of data points that we actually observe.It's our sample size.So knowing the degrees of freedomallows us to determine the shape of the t-distribution.Here are two examples, one which has degrees

  • 04:54

    DANIEL LITTLE [continued]: of freedom of only one-- so are our samplesize in that particular case wouldbe a sample of two people or two observations--and another which has a degrees of freedom of 1,000,so our total sample size would be 1,001 people.You can see that in both of these casesthe t-distribution is symmetric and bell

  • 05:14

    DANIEL LITTLE [continued]: shaped just like the normal distributionbut has heavier tails.What this means is that it's moreprone to producing values which fallfar away from the mean value.As the degrees of freedom increase,the distribution becomes steeper.So it starts out being very shallow.You could see the shallow distribution here,

  • 05:35

    DANIEL LITTLE [continued]: but with an increase of degrees of freedom,the distribution actually becomes quite steep,meaning that more extreme values becomeless probable whenever our degrees of freedom are large.So using this t-distribution, we candetermine the probability of our observed t-statistic value.

  • 05:57

    DANIEL LITTLE [continued]: Our null hypothesis is that t equals 0, so what we're doingis determining the probability that our observed tvalue, given the null hypothesis assumption that t equals0, what we're attempting to discernis the p value of that distribution.You can find the probabilities specified by the t-distribution

  • 06:21

    DANIEL LITTLE [continued]: by finding the area under the curvejust like you can for the normal distribution,or you can use tables which are commonlyavailable in the back of statistics textbooksor by using a computer program to actually compute those pvalues.

  • 06:41

    DANIEL LITTLE [continued]: Once you have the p value, what we want to knowis whether or not our observed t valuecomes from the tails of the t-distribution.Is that p value less than 0.05?If the p value is less than 0.05,then what we would conclude is that our observed t valuewas unlikely to have been generated by a t-distribution

  • 07:05

    DANIEL LITTLE [continued]: with a mean of 0.Consequently, our observed sampleof data, our single sample of datais unlikely to have been generated by the populationthat we thought it was generated from.To summarize, when we have a single sample of data,we make some hypothesis about the mean of the population.We then use the t-statistic and associated p value

  • 07:26

    DANIEL LITTLE [continued]: to determine the probability of our observed data giventhis hypothesis.In the next video, I will discussusing the t-test for a case when you have two samples of data.

Video Info

Series Name: Statistics for Psychology

Episode: 12

Publisher: University of Melbourne

Publication Year: 2014

Video Type:Tutorial

Methods: T-test, Hypothesis testing, Degrees of freedom, P-value

Keywords: mathematical concepts

Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:



In chapter 12 of his series of statistics for psychology, Professor Daniel Little begins a three-part section on t-tests. Little discusses z-values and analysis using a single sample of data.

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T-test I: The z-test and the single sample t-test

In chapter 12 of his series of statistics for psychology, Professor Daniel Little begins a three-part section on t-tests. Little discusses z-values and analysis using a single sample of data.

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