Chapter 8Single Sample Tests

Part II: Introduction to Hypothesis Testing

Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences

Single Sample z-statistic

Mean Score Formula

z = X

X

X

Mean Score Formula Simplified

z = XX

Figure 8.1

When do we use the z-test?

1. When the experiment involves a single sample mean and the parameters of the corresponding null hypothesis population are known (μ and σ).

When do we use the z-test?

2. When the sampling distribution is normally distributed, which is the case if:

a. n ≥ 30 or

b. The null hypothesis population of raw scores is known to be normally distributed.

Single Sample t test

Comparison of Single Sample z and Single Sample t Formulas

z = XX =

n

X

t = XsX =

sn

X

Single Sample t testThe t critical values are dependent on sample size

(n) Now estimating from s, and the accuracy of σ

that estimation is dependent on n.Our critical probability values are now going to vary

with our sample size, unlike the z distribution in which the probabilities were independent of sample size.  

Single Sample t testNew distribution called the t distribution.

It will vary with degrees of freedom (df), which are

related to sample size.

Single Sample t testDegrees of freedom: The number of scores that are

“free to vary” when calculating a statistic. The remaining value or values are then fixed.

Single Sample t testScore (X) Sample Mean ( ) Deviation

3 6 -3

5 6 -1

10 6 FIXED VALUE = 4

  ∑ Deviations = 0

X

18X

Figure 8.2Flowchart for choosing the appropriate test

Confidence IntervalsConfidence interval for the population mean: Range of

values with a calculated probability of containing the mean

Confidence IntervalsConfidence limits for the population mean: The upper

and lower values (or boundaries) surrounding the confidence interval.

Figure 8.3Graphic Display of a 95% Confidence Interval

Confidence IntervalsConfidence limits for the population mean: The upper

and lower values (or boundaries) surrounding the confidence interval.

μLower = X - s X (t 0.025 ) μUpper = X + s X (t 0.025 )

Effect Sizes and PowerEffect size is a standardized measure of the difference

between two (or more) group means; it is the difference in means divided by the shared standard deviation of two or more groups.

Effect Sizes and PowerFor a population with a known standard deviation ( ), σ

you can use Cohen’s d to calculate the effect size.

Then you can plug your effect size into a software package to calculate power.

Cohen’s D

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