# PSY201H1 Lecture Notes - Lecture 9: Variance, Standard Error, Test Statistic

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PSY201 Lecture 9; Nov. 17, 2011

Introduction to the t Statistic

The t Statistic: An Alternative to z

•When hypothesis testing:

○Use a sample mean M to approximate the

pop mean µ.

○Standard error, σM, measures how well M

approximates µ

σM = σ/√n

○Then compare M to µ by computing the

relevant z-score test statistic

z = (M - µ)/ σM

Use to determine whether the obtained

dif is greater than expected by chance

by looking at unit normal table (for a

normal distribution)

•Like z-score, t statistic allows researchers to use

sample data to test hypotheses about an

unknown µ

•Particular advantage of t statistic:

○Doesn’t require any knowledge of the pop

σwhich we typically don’t know

○So can be used to test hypotheses about a

completely unknown pop, ie:

Both µ & σ unknown

Only available info about pop comes

from sample

•All required for a hypothesis test with t is a

sample & a reasonable hypothesis about µ

•When we don’t haveσcan estimate it using the

sample variability

○Much like estimating µ using M

•Can use t statistic when determining whether

treatment causes a change in µ

•Sample is obtained from the pop treatment is

administered to sample

○As usual, if resulting sample M is significantly

different from original µ, can conclude that

treatment has a significant effect

•Like before, hypothesis test attempts to decide

btwn:

○Ho: Is it reasonable that the discrepancy

btwn M & µ is simply due to sampling error

and not result of treatment effect?

○H1: Is the discrepancy btwn M & µ more than

expected by sampling error alone?

ie. Is M significantly different from µ

•Critical 1st step for t statistic hypothesis test:

○Calculate exactly how much dif btwn M & µ is

reasonable to expect

○But since σ is unknown, impossible to

compute standard error of M (σM = σ/√n) as

done w z-scores

○ t statistic requires to use sample

data’s variance, s2 to compute

estimated standard error, sM

• sM = s/√n

○Rmbr s2 = SS/(n-1)

○ s = √(SS/df)

•sM used more in terms of sample variance, since

provides accurate & unbiased estimate of the

pop variance σ2

○ Estimated sample error = =

○Rmbr must know sample mean before

computing sample variance, so there is a

restriction on sample variability

○Only n - 1 scores are independent & free to

vary

○ n - 1 = degrees of freedom (df) for sample

variance

•t statistic (like the z-score) forms a ratio:

○Numerator: Obtained dif btwn M

hypothesized µ

○Denominator: Estimated standard error

(measures how much dif expected by

chance)

•A large value for t (a large ratio) indicates

obtained difference btwn data & hypothesis >

expected if treatment has no effect (ie. is just

error)

○W large sample, df large, so estimation is

very good

So t statistic will be very similar to z-

score

○W small samples, df small, so t statistic

provides relatively poor estimate of z

•Can think of the t statistic as "estimated zscore."

;

•Just like we can make a distribution of z-scores,

we can make a t distr

○Complete set of t values computed for every

possible random sample for a specific

sample size n, or specific degrees of freedom

df

•As df approaches infinity, t distr

approximates normal distr

•How well it approximates normal distr

depends on df

○ie. There’s a “family” of t distrs so there’s a

dif sampling distr of t for each possible df

•Larger n larger n – 1 better the t distr

approximates the normal distr

•W small values for df t distr flatter & more

spread out than a normal distr