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### PSY 307 – Statistics for the Behavioral Sciences

Chapter 11-12 – Confidence Intervals, Effect Size, Power

Point Estimates

- The best estimate of a population mean is the sample mean.
- When we use a sample to estimate parameters of the population, it is called a point estimate.
- How accurate is our point estimate?
- The sampling distribution of the mean is used to evaluate this.

Confidence Interval

- The range around the sample mean within which the true population mean is likely to be found.
- It consists of a range of values.
- The upper and lower values are the confidence limits.
- The range is determined by how confident you wish to be that the true mean falls between the values.

What is a Confidence Interval?

- A confidence interval for the mean is based on three elements:
- The value of the statistic (e.g., the mean, m).
- The standard error (SE) of the measure (sx).
- The desired width of the confidence interval (e.g., 95% or 99%, 1.96 for z).
- To calculate for z: m ± (zconf)(sx)

Levels of Confidence

- A 95% confidence interval means that if a series of confidence intervals were constructed around different means, about 95% of them would include the true population mean.
- When you use 99% as your confidence interval, then 99% would include the true pop mean.

Demos

- http://www.stat.sc.edu/~west/javahtml/ConfidenceInterval.html
- http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/

Calculating Different Levels

- For 95% use the critical values for z scores that cutoff 5% in the tails:
- 533 ± (1.96)(11) = 554.56 & 511.44

where M = 533 and sM = 11

- For 99% use the critical values that cutoff 1% in the tails:
- 533 ± (2.58)(11) = 561.38 & 504.62

Sample Size

- Increasing the sample size decreases the variability of the sampling distribution of the mean:

Effect of Sample Size

- Because larger sample sizes produce a smaller standard error of the mean:
- The larger the sample size, the narrower and more precise the confidence interval will be.
- Sample size for a confidence interval, unlike a hypothesis test, can never be too large.

Other Confidence Intervals

- Confidence intervals can be calculated for a variety of statistics, including r and variance.
- Later in the course we will calculate confidence intervals for t and for differences between means.
- Confidence intervals for percents or proportions frequently appear as the margin of error of a poll.

Effect Size

- Effect size is a measure of the difference between two populations.
- One population is the null population assumed by the null hypothesis.
- The other population is the population to which the sample belongs.
- For easy comparison, this difference is converted to a z-score by dividing it by the pop std deviation, s.

Calculating Effect Size

- Subtract the means and divide by the null population std deviation:
- Interpreting Cohen’s d:
- Small = .20
- Medium = .50
- Large = .80

Comparisons Across Studies

- The main value of calculating an effect size is when comparing across studies.
- Meta-analysis – a formal method for combining and analyzing the results of multiple studies.
- Samples sizes vary and affect significance in hypothesis tests, so test statistics (z, t, F) cannot be compared.

Probabilities of Error

- Probability of a Type I error is a.
- Most of the time a = .05
- A correct decision exists .95 of the time (1 - .05 = .95).
- Probability of a Type II error is b.
- When there is a large effect, b is very small.
- When there is a small effect, b can be large, making a Type II error likely.

When there is no effect…

a = .05

Sample means that produce a type I error

Hypothesized and true distributions coincide

.05

COMMON

1.65

Effect Size and Distribution Overlap

- Cohen’s d is a measure of effect size.
- The bigger the d, the bigger the difference in the means.
- http://www.bolderstats.com/gallery/normal/cohenD.html

Power

- The probability of producing a statistically significant result if the alternative hypothesis (H1) is true.
- Ability to detect an effect.
- 1- b (where b is the probability of making a Type II error)

Calculating Power

- Most researchers use special purpose software or internet power calculators to determine power.
- This requires input of:
- Population mean, sample mean
- Population standard deviation
- Sample size
- Significance level, 1 or 2-tailed test
- http://www.stat.ubc.ca/~rollin/stats/ssize/n2.html

How Power Changes with N

- WISE Demo
- http://wise.cgu.edu/powermod/exercise1b.asp

Effect of Larger Sample Size

Larger samples produce smaller standard deviations.

Smaller standard deviations mean less overlap between two distributions.

b Decreases with Larger N’s

Note: This is for an effect in the negative direction (H0 is the red curve on the right).

Increasing Power

- Strengthen the effect by changing your manipulation (how the study is done).
- Decrease the population’s standard deviation by decreasing noise and error (do the study well, use a within subject design).
- Increase sample size.
- Change the significance level.

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