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Monday, October 21. Hypothesis testing using the normal Z-distribution. Student’s t distribution. Confidence intervals. An Example. You draw a sample of 25 adopted children. You are interested in whether they are different from the general population on an IQ test (  = 100,  = 15).

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Monday, October 21

Hypothesis testing using the normal Z-distribution.

Student’s t distribution.

Confidence intervals.


An Example

You draw a sample of 25 adopted children. You are interested in whether they

are different from the general population on an IQ test ( = 100,  = 15).

The mean from your sample is 108. What is the null hypothesis?

H0:  = 100

Test this hypothesis at  = .05

Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean that

differs from  by an amount as large or larger than what was observed.

Step 4. Make a decision regarding H0, whether to reject or not to reject it.


Step 1. State the statistical hypothesis H0 to be tested (e.g., H0:  = 100)

Step 2. Specify the degree of risk of a type-I error, that is, the risk of incorrectly concluding

that H0 is false when it is true. This risk, stated as a probability, is denoted by , the probability

of a Type I error.

Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean that

differs from  by an amount as large or larger than what was observed.

Step 4. Make a decision regarding H0, whether to reject or not to reject it.


Step 1. State the statistical hypothesis H0 to be tested (e.g., H0:  = 100)

Step 2. Specify the degree of risk of a type-I error, that is, the risk of incorrectly concluding

that H0 is false when it is true. This risk, stated as a probability, is denoted by , the probability

of a Type I error.

Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean that

differs from  by an amount as large or larger than what was observed, find the critical values of an observed sample mean whose deviation from 0 would be “unlikely”, defined as a probability < .

Step 4. Make a decision regarding H0, whether to reject or not to reject it,



X -

X - 

_

_

z =

t =

-

-

X

sX

s

-

sX =

 N


The t-distribution is a family of distributions varying by degrees of freedom (d.f., where

d.f.=n-1). At d.f. =, but at smaller than that, the tails are fatter.


Degrees of Freedom degrees of freedom (

df = N - 1


Problem degrees of freedom (

Sample:

Mean = 54.2

SD = 2.4

N = 16

Do you think that this sample could have been drawn from a population with  = 50?


X - degrees of freedom (

t =

-

sX

Problem

Sample:

Mean = 54.2

SD = 2.4

N = 16

Do you think that this sample could have been drawn from a population with  = 50?

_


The mean for the sample of 54.2 (sd = 2.4) was significantly different from a hypothesized population mean of 50, t(15) = 7.0, p < .001.


The mean for the sample of 54.2 (sd = 2.4) was significantly reliably different from a hypothesized population mean of 50, t(15) = 7.0, p < .001.


Interval Estimation (a.k.a. confidence interval)

Is there a range of possible values for  that you can specify, onto which you can attach a statistical probability?


Confidence Interval

_

_

X - tsX   X + tsX

Where

t = critical value of t for df = N - 1, two-tailed

X = observed value of the sample

_


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