Making estimations Statistics for the Social Sciences Psychology 340 Spring 2010.

Making estimations

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social Sciences

Statistical analysis follows design

• Are you looking for a difference between groups?

• Are you estimating the mean (or a mean difference)?

• Are you looking for a relationship between two variables?


Social SciencesEstimation

• So far we’ve been dealing with situations where we know the population mean. However, most of the time we don’t know it.

μ = ?

• Two kinds of estimation– Point estimates

• A single score

– Interval estimates• A range of scores



μ = ?

Two kinds of estimation– Point estimates

– Interval estimates

Advantage Disadvantage

A single score

A range of scoresConfidence of the estimate

Little confidence of the estimate “the mean is 85”

“the mean is somewhere between 81 and 89”



• Both kinds of estimates use the same basic procedure– The formula is a variation of the test statistic formula (so far we

know the z-score)

zX

X

X

X

zX

(X

) X X

€

X

= X ± zX

(σX

)




know the z-score)

1) It is often the only piece of evidence that we have, so it is our best guess.2) Most sample means will be pretty close to the population mean, so we have a good chance that our sample mean is close.

Why the sample mean?

€

X

= X ± zX

(σX

)




know the z-score)

1) A test statistic value (e.g., a z-score)2) The standard error (the difference that you’d expect by chance)

Margin of error

€

X

= X ± zX

(σX

)



– Step 1: You begin by making a reasonable estimation of what the z (or t) value should be for your estimate.

• For a point estimation, you want what? z (or t) = 0, right in the middle

• For an interval, your values will depend on how confident you want to be in your estimate

– What do I mean by “confident”?

» 90% confidence means that 90% of confidence interval estimates of this sample size will include the actual population mean

€

X

= X ± zX

(σX

)

• Both kinds of estimates use the same basic procedure



– Step 1: You begin by making a reasonable estimation of what the z (or t) value should be for your estimate.

• For a point estimation, you want what? z (or t) = 0, right in the middle

• For an interval, your values will depend on how confident you want to be in your estimate

– Step 2: You take your “reasonable” estimate for your test statistic, and put it into the formula and solve for the unknown population

parameter.

€

X

= X ± zX

(σX

)

• Both kinds of estimates use the same basic procedure


Social Sciences Estimates with z-scores

Make a point estimate of the population mean given a sample with a X = 85, n = 25, and a population σ = 5.

X

X zX

(X

)

85 (0)5

25

85 So the point estimate is the sample mean



Make an interval estimate with 95% confidence of the population mean given a sample with a X = 85, n = 25, and a population σ = 5.

X

X zX

(X

)

-1-2 1 2

95%

What two z-scores do 95% of the data lie between?





So the confidence interval is: 83.04 to 86.96

X

X zX

(X

)

85 (1.96)5

25

86.96

From the table: z(1.96) =.0250

-1-2 1 2

95%

2.5% 2.5%

83.04

or 85 ± 1.96






X

X zX

(X

)

85 (1.65)5

25

86.65


83.35

or 85 ± 1.65

-1-2 1 2 -1-2 1 2

5% 5%

90%






X

X zX

(X

)

85 (1.65)5

4

89.13


80.88

or 85 ± 4.13

-1-2 1 2 -1-2 1 2

5% 5%

90%


Social Sciences Estimation in other designs

Confidence interval

sX =sn

Diff. Expected by chance

X = X ± (tcrit )(sX )

Estimating the mean of the population from one sample, but we don’t know the σ

How do we find this?

How do we find this?

Use the t-table


Social Sciences Estimates with t-scores

Proportion in one tail0.10 0.05 0.025 0.01 0.005

Proportion in two tailsdf 0.20 0.10 0.05 0.02 0.01: : : : : :5 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707: : : : : :

-1-2 1 2

Confidence intervals always involve + a margin of errorThis is similar to a two-tailed test, so in the t-table, always use the “proportion in two tails” heading, and select the α-level corresponding to (1 - Confidence level)

What is the tcrit needed for a 95% confidence interval?

95%

95% in middle

2.5% 2.5%

so two tails with 2.5% in each

2.5%+2.5% = 5% or α = 0.05, so look here


Social Sciences

Make an interval estimate with 95% confidence of the population mean given a sample with a X = 85, n = 25, and a sample s = 5.

Estimates with t-scores

What two critical t-scores do 95% of the data lie between?


X = X ± tcrit (sX )85 ± (2.064)5

25

⎛⎝⎜

⎞⎠⎟

87.06

From the table:

tcrit =+2.064

-1-2 1 2

95%

2.5% 2.5%

82.94

or 85 ± 2.064

df n−125 −124

95% confidence

Proportion in one tail 0.10 0.05 0.025 0.01 0.005

Proportion in two tails df 0.20 0.10 0.05 0.02 0.01 : : : : : :

24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 : : : : : :



€

sD

=sD

nD

Confidence interval

Diff. Expected by chance €

D

= D ± (tcrit )(sD

)

Estimating the difference between two population means based on two related samples



Confidence interval

€

A − μB = (X A − X B ) ± (tcrit )(sX A −X B

)

Estimating the difference between two population means based on two independent samples

sXA −XB=

sP2

nA

+sP

2

nB

Diff. Expected by chance


Social Sciences Estimation Summary

Design Estimation (Estimated) Standard error

€

A − μB = (X A − X B ) ± (tcrit )(sX A −X B

) sXA −XB=

sP2

nA

+sP

2

nB€

sD

=sD

nD

€

D

= D ± (tcrit )(sD

)

sX =sn

X = X ± (tcrit )(sX )

€

X

= X ± zX

(σX

) X =σ

nOne sample, σ known

One sample, σ unknown

Two related samples, σ unknown

Two independent samples, σ unknown


Social Sciences

Statistical analysis follows design

• Questions to answer:• Are you looking for a

difference, or estimating a mean?

• Do you know the pop. SD (σ)?

• How many samples of scores?

• How many scores per participant?

• If 2 groups of scores, are the groups independent or related?

• Are the predictions specific enough for a 1-tailed test?


Social Sciences Design Summary

DesignOne sample, σ known, 1 score per sub

One sample, σ unknown, 1 score per

2 related samples, σ unknown, 1 score per - or –1 sample, 2 scores per sub, σ unknown

Two independent samples, σ unknown,1 score per sub

Independent samples-t

€

A − μB = (X A − X B ) ± (tcrit )(sX A −X B)

t =(XA −XB)−(A −B)

sXA −XB

sXA −XB=

sP2

nA

+sP

2

nB

Related samples t

€

D

= D ± (tcrit )(sD

)t =

D−D

sD

€

sD

=sD

nD

One sample t

X = X ± (tcrit )(sX )t =X −X

sX

sX =sn

One sample z

€

X

= X ± zX

(σX

)zX =X −X

X

X =σ

n










Researchers used a sample of n = 16 adults. Each person’s mood was rated while smiling and frowning. It was predicted that moods would be rated as more positive if smiling than frowning. Results showed Msmile = 7 and Mfrown = 4.5.

Are the groups different?








Researcher measures reaction time for n = 36 participants. Each is then given a medicine and reaction time is measured again. For this sample, the average difference was 24 ms, with a SD of 8. With 95% confidence estimate the population mean difference.

A teacher is evaluating the effectiveness of a new way of presenting material to students. A sample of 16 students is presented the material in the new way and are then given a test on that material, they have a mean of 87. How do they compare to past

classes with a mean of 82 and SD = 3?

t =D−D

sD

Related samples t

Related samples CI

€

D

= D ± (tcrit )(sD

)

1 sample z zX =X −X

X

Making estimations Statistics for the Social Sciences Psychology 340 Spring 2010.

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