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 (σ X )




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 (σ X )




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 (σ X )



– 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 (σ X )

• 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 (σ X )

• 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) 525

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 )

12 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) 525

86.96

From the table: z(1.96) =.0250

12 1 2

95%

2.5% 2.5%

83.04

or 85 ± 1.96






X X zX ( X )

85 (1.65) 525

86.65


83.35

or 85 ± 1.65

12 1 2 12 1 2

5% 5%

90%






X X zX ( X )

85 (1.65) 54

89.13


80.88

or 85 ± 4.13

12 1 2 12 1 2

5% 5%

90%


Social Sciences Estimation in other designs

Confidence interval

sX n

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: : : : : :

12 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)525

⎛⎝⎜

⎞⎠⎟

87.06

From the table:

tcrit =+2.064

12 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

P2

nA

+P

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

P

2

nA

+P

2

nB€

sD = sD

nD

€


sX n

X = X ± (tcrit )(sX )

€

X = X ± zX (σ X ) X =σn

One 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)

XA−XB

sXA −XB

P2

nA

+P

2

nB

Related samples t

€

D = D ± (tcrit )(sD )t

D −D

D

€

sD = sD

nD

One sample t

X = X ± (tcrit )(sX )t X −X

X

sX n

One sample z

€

X = X ± zX (σ X )zX X −X

X

X =σn










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

D

Related samples t

Related samples CI

€


1 sample z zX X −X

X

psychology 340 spring 2010

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