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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 .

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### PSY 307 – Statistics for the Behavioral Sciences

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

• 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.

• 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.

• 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)

• 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.

• http://www.stat.sc.edu/~west/javahtml/ConfidenceInterval.html

• http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/

• 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

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

• 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.

• 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 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.

Effect Size

X1

X2

Effect Size

X1

X2

Critical Value

Critical Value

• Subtract the means and divide by the null population std deviation:

• Interpreting Cohen’s d:

• Small = .20

• Medium = .50

• Large = .80

• 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.

• 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.

a = .05

Sample means that produce a type I error

Hypothesized and true distributions coincide

.05

COMMON

1.65

• 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

• 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)

Power

Effect Size

X1

X2

Critical value

Power

Effect Size

X1

X2

Critical Value

Critical Value

• 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

• WISE Demo

• http://wise.cgu.edu/powermod/exercise1b.asp

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).

• 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.