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Amusing Statistics

• Use what you know about normal distributions to evaluate this finding:

The study, published in Pediatrics, the journal of the American Academy of Pediatrics, found that among the 4,508 students in Grades 5-8 ﾊ who participated, 36 per cent reported excellent school performance, 38 per cent reported good performance, 20 per cent said they were average performers, and 7 per cent said they performed below average.

Review

• The Z-test is used to compare the mean of a sample to the mean of a population

€

Zx =x − μ x

σ x

€

σX

=σ

nand

Review

• The Z-score is normally distributed

Review

• The Z-score is normally distributed

• Thus the probability of obtaining any given Z-score by random sampling is given by the Z table

Review

• We can likewise determine critical values for Z such that we would reject the null hypothesis if our computed Z-score exceeds these values– For alpha = .05:

• Zcrit (one-tailed) = 1.64• Zcrit (two-tailed) = 1.96

Confidence Intervals

• A related question you might ask:– Suppose you’ve measured a mean and

computed a standard error of that mean

– What is the range of values such that there is a 95% chance of the population mean falling within that range?

Gaussian (Normal) Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

score

probability

• There is a 2.5% chance that the population mean is actually 1.96 standard errors more than the observed mean


95%1.96

2.5%

True mean?

Gaussian (Normal) Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

score

probability

• There is a 2.5% chance that the population mean is actually 1.96 standard errors less than the observed mean


2.5%-1.96

95%

True mean?

• Thus there is a 95% chance that the true population mean falls within + or - 1.96 standard errors from a sample mean


• Thus there is a 95% chance that the true population mean falls within + or - 1.96 standard errors from a sample mean

• Likewise, there is a 95% chance that the true population mean falls within + or - 1.96 standard deviations from a single measurement


• This is called the 95% confidence interval…and it is very useful

• It works like significance bounds…if the 95% C.I. doesn’t include the mean of a population you’re comparing your sample to, then your sample is significantly different from that population


• Consider an example:

• You measure the concentration of mercury in your backyard to be .009 mg/kg

• The concentration of mercury in the Earth’s crust is .007 mg/kg. Let’s pretend that, when measured at many sites around the globe, the standard deviation is known to be .002 mg/kg


• The 95% confidence interval for this mercury measurement is


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backyard = .009mg /kg

€

σ =.002mg /kg

€

95%C.I. = x + /− Zcrit (two − tailed) ×σ

€

=.009 + /−1.96 × .002mg /kg

€

=.0051 → .0129

• This interval includes .007 mg/kg which, it turns out, is the mean concentration found in the earth’s crust in general

• Thus you would conclude that your backyard isn’t artificially contaminated by mercury


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.0051 ≥ .007 ≥ .0129

• Imagine you take 25 samples from around Alberta and you found:


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x = .009mg /kg

€

σ =.002mg /kg

€

σx

=σ

n=

.002

25= .0004

• Imagine you take 25 samples from around Alberta and you found:

• .009 +/- (1.96 x .0004) = .008216 to .009784

• This interval doesn’t include the .007 mg/kg value for the earth’s crust so you would conclude that Alberta has an artificially elevated amount of mercury in the soil


Power

• we perform a Z-test and determine that the difference between the mean of our sample and the mean of the population is not due to chance with a p < .05

Power


• we say that we have a significant result…

Power


• we say that we have a significant result…

• but what if p is > .05?

Power

• What are the two reasons why p comes out greater than .05?

Power

• What are the two reasons why p comes out greater than .05?

– Your experiment lacked Statistical Power and you made a Type II Error

– The null hypothesis really is true

Power

• Two approaches:– The Hopelessly Jaded Grad Student

Solution

– The Wise and Well Adjusted Professor Procedure

Power

1. Hopelessly Jaded Grad Student Solution - conclude that your hypothesis was wrong and go directly to the grad student pub

Power

- This is not the recommended course of action

Power

2. The Wise Professor Procedure - consider the several reasons why you might not have detected a significant effect

Power

- recommended by wise professors the world over

Power

• Why might p be greater than .05 ?

• Recall that:

€

Zx =x − μ x

σ x

€

σX

=σ

nand

Power• Why might p be greater than .05 ?

1. Small effect size:

– The effect doesn’t stand out from the variability in the data– You might be able to increase your effect size (e.g. with a larger dose or treatment)

€

X is quite close to the mean of the population


2. Noisy Data

– A large denominator will swamp the small effect– Take greater care to reduce measurement errors

€

σand therefore

€

σX is quite large


3. Sample Size is Too Small

– A large denominator will swamp the small effect – Run more subjects

€

σX is quite large because

€

n is small

Power

• The solution in each case is more power:

Power• The solution in each case is more power:• Power is like sensitivity - the ability to detect small effects in noisy data

Power• The solution in each case is more power:• Power is like sensitivity - the ability to detect small effects in noisy data• It is the opposite of Type II Error rate

Power• The solution in each case is more power:• Power is like sensitivity - the ability to detect small effects in noisy data• It is the opposite of Type II Error rate• So that you know: there are equations for computing statistical power

Power

• An important point about power and the null hypothesis:

– Failing to reject the null hypothesis DOES NOT PROVE it to be true!!!

Power


– How to prove that smoking does not cause cancer:

• enroll 2 people who smoke infrequently and use an antique X-Ray camera to look for cancer

• Compare the mean cancer rate in your group (which will probably be zero) to the cancer rate in the population (which won’t be) with a Z-test

Power


– If p came out greater than .05, you still wouldn’t believe that smoking doesn’t cause cancer

Power


– If p came out greater than .05, you still wouldn’t believe that smoking doesn’t cause cancer

– You will, however, often encounter statements such as “The study failed to find…” misinterpreted as “The study proved no effect of…”

• We’ve been using examples in which a single sample is compared to a population

Experimental Design


• Often we employ more sophisticated designs

Experimental Design


• Often we employ more sophisticated designs

• What are some different ways you could run an experiment?

Experimental Design

Experimental Design

• Compare one mean to some value– Often that value is zero

Experimental Design

• Compare one mean to some value– Often that value is zero

• Compare two means to each other

Experimental Design

• There are two general categories of comparing two (or more) means with each other

Experimental Design

1. Repeated Measures - also called “within-subjects” comparison

• The same subjects are given pre- and post- measurements

• e.g. before and after taking a drug to lower blood pressure

• Powerful because variability between subjects is factored out

• Note that pre- and post- scores are linked - we say that they are dependant

• Note also that you could have multiple tests

Experimental Design

1. Problems with Repeated-Measure design:

• Practice/Temporal effect - subjects get better/worse over time

• The act of measuring might preclude further measurement - e.g. measuring brain size via surgery

• Practice effect - subjects improve with repeated exposure to a procedure

Experimental Design

2. Between-Subjects Design• Subjects are randomly assigned to treatment

groups - e.g. drug and placebo• Measurements are assumed to be statistically

independent

Experimental Design

2. Problems with Between-Subjects design

• Can be less powerful because variability between two groups of different subjects can look like a treatment effect

• Often needs more subjects

Experimental Design

• We’ll need some statistical tests that can compare:

– One sample mean to a fixed value– Two dependent sample means to each

other (within-subject)– Two independent sample means to each

other (between-subject)

Experimental Design

• The t-test can perform each of these functions

• It also gets around a big problem with the z-test…

Problems with Z

and what to do instead

The Z statistic

• The Z statistic (with which to compare to the Zcrit)

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

• What is the problem you will encounter in trying to use this statistic?

The Z statistic

• What is the problem you will encounter in trying to use this statistic?

• Although you might have a guess about the population mean, you will almost certainly not know the population variance!

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

• What to do?

• Could we estimate

• What would we use and what would have to be the case for it to be useful?

€

σ 2

The Z statistic

• What to do?

• Could we estimate

• What would we use and what would have to be the case for it to be useful?

• We could use our sample variance, S2 to estimate the population variance

€

σ 2

€

σ 2

Estimating Population Variance

• Just like there are many sample means (the sampling distribution of the mean) there are many S2s



• tends to be near the value of but does S2 tend to be near the value of

€

X

€

μ

€

σ 2



• tends to be near the value of but does S2 tend to be near the value of

• No. It is a biased estimator. It tends to be lower than €

X

€

σ 2

€

μ

€

σ 2


• Why is S2 biased?



• The sum of the deviation scores in your sample must equal zero regardless of where they came from in the population




• This means that the deviations in your sample are somewhat more constrained than in the population




• This means that the deviations in your sample are somewhat more constrained than in the population

• S2 is has relatively fewer degrees of freedom than the entire population


• Specifically S2 has n - 1 degrees of freedom


• Specifically S2 has n - 1 degrees of freedom

• So if we compute S2 but use n - 1 instead of n in the denominator we’ll get an unbiased estimator of

€

σ 2


• Of course if you’ve already computed S2 using n in the denominator you can multiply by n to recover the sum of squared deviations and then divide by n-1

The t Statistic(s)

• Using an estimated , which we’ll call we can create an estimate of which we’ll call

€

ˆ σ X

=ˆ σ

n€

σ 2

€

ˆ σ 2

€

σX

€

ˆ σ X

where

€

ˆ σ =(X i − X )2

n −1∑ =

nS2

n −1

The t Statistic(s)

• Using, instead of we get a statistic that isn’t from a normal (Z) distribution - it is from a family of distributions called t

€

tn−1 =x − μ x

ˆ σ x

€

ˆ σ X

€

σX

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