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

2. 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?

3. Estimation
So far we’ve been dealing with situations where we know the population mean. However, most of the time we don’t know it.
Two kinds of estimation
Point estimates
A single score
Interval estimates
A range of scores μ
= ?

4. Estimation μ = ? Advantage Disadvantage Two kinds of estimation
Point estimates
Interval estimates
Confidence of the estimate
Little confidence of the estimate
A single score
A range of scores
“the mean is 85”
“the mean is somewhere between 81 and 89”

5. Estimation 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)

6. Estimation 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)
Why the sample mean?
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.

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

8. Estimation 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
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

9. Estimation 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.

10. 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.
So the point estimate is the sample mean

11. Estimates with z-scores
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.
What two z-scores do 95% of the data lie between?
95%

12. Estimates with z-scores
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.
What two z-scores do 95% of the data lie between?
From the table:
z(1.96) =.0250
So the confidence interval is: to 86.96
or 85 ± 1.96
2.5%
95%

13. Estimates with z-scores
Make an interval estimate with 90% confidence of the population mean given a sample with a X = 85, n = 25, and a population σ = 5.
What two z-scores do 90% of the data lie between?
From the table:
z(1.65) =.0500
So the confidence interval is: to 86.65
or 85 ± 1.65 5%
90%

14. Estimates with z-scores
Make an interval estimate with 90% confidence of the population mean given a sample with a X = 85, n = 4, and a population σ = 5.
What two z-scores do 90% of the data lie between?
From the table:
z(1.65) =.0500
So the confidence interval is: to 89.13
or 85 ± 4.13 5%
90%

15. Estimation in other designs
Estimating the mean of the population from one sample, but we don’t know the σ
How do we find this?
Use the t-table
Confidence interval
How do we find this?
Diff. Expected by chance

16. Estimates with t-scores
Confidence intervals always involve + a margin of error
This 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?
2.5%
so two tails with 2.5% in each
2.5%+2.5% = 5% or α = 0.05, so look here
95%
95% in middle

17. Estimates with t-scores
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.
What two critical t-scores do 95% of the data lie between?
From the table:
tcrit =+2.064
So the confidence interval is: to 87.06
95% confidence
or 85 ± 2.064
2.5%
95%

18. Estimation in other designs
Estimating the difference between two population means based on two related samples
Confidence interval
Diff. Expected by chance

19. Estimation in other designs
Estimating the difference between two population means based on two independent samples
Confidence interval
Diff. Expected by chance

20. Estimation Summary Design Estimation (Estimated) Standard error
One sample, σ known
One sample, σ unknown
Two related samples, σ unknown
Two independent samples, σ unknown

21. 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?

22. Design Summary 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?
Design
One sample, σ known, 1 score per sub
One sample z
One sample, σ unknown, 1 score per
One sample t
2 related samples, σ unknown, 1 score per
- or –
1 sample, 2 scores per sub, σ unknown
Related samples t
Independent samples-t
Two independent samples, σ unknown,
1 score per sub

23. Estimates with z-scores
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?
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?
Related samples t
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.
Related samples CI
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?
1 sample z