Psy 307 statistics for the behavioral sciences
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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

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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.




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.

Effect Size

Effect Size



A Significant Effect

Effect Size



Critical Value

Critical Value

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




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.



  • 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)

Small Effects Have Low Power


Effect Size



Critical value

Large Effects Have More Power


Effect Size



Critical Value

Critical Value

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


Sample Power Graph 1

Sample Power Graph 2

How Power Changes with N

  • WISE Demo


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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