t-Tests give us more options

Inferential

Statistics

What we know already:

Frequency distributions shown in graphs

Summarizing data sets

Central tendency (mean, median, mode)

Variability (variance and standard deviation)

Central Limit Theorem

Sample means are normally distributed

If n > 30 or population is normally distributed

Hypothesis Testing basics

Null hypothesis & critical region defined

Sample data used to make a decision

Bad news:

“The shortcoming of using a z-score as an inferential statistic is that the z-score formula requires more information than is usually available.”

“Specifically, a z-score requires that we know the value of the population standard deviation (or variance). In most situations, however, the standard deviation for the population is not known.”

One-Sample t-Test

We want to compare a single sample to a

population mean, or a hypothetical mean

We don’t know σ2 so we use s2, the Sample Variance

We have an estimated standard error, sM

We use sM to compute t t is similar to z in shape: bell curve

It is shorter in the center, and the tails have more area

There is more than one t curve The curve is determined by M, s2, and df

Degrees of freedom, df is related to sample size

Distribution of the t statistic for different values of degrees of freedom are

compared to a normal z-score distribution. Like the normal distribution, t

distributions are bell-shaped and symmetrical and have a mean of zero. However, t

distributions have more variability, indicated by the flatter and more spread-out

shape. The larger the value of df is, the more closely the t distribution approximates

a normal distribution.

One-Sample t-Test

The same 4 steps of hypothesis testing

State the hypotheses, choose the alpha level

Locate the critical region on the table of t values

Collect sample data and compute t statistic

Evaluate H0 and make a decision

We can measure effect size with Cohen’s d and

with r 2 (proportion of variability explained)

We can have directional (one-tailed) t-tests

Comparison of z- and t-Tests

One-sample z-test

Sample is drawn from a

population whose mean is

known or hypothesized

σ2 is known, compute σM

z=(M-µ) / σM

One sample t-test

Sample is drawn from a

population whose mean is

known or hypothesized

σ2 not known

Use s2 to compute sM

df = n-1

t = (M-µ) / sM

2

Mn

2

2 so M

SSSS s df

s sdf n n

One-sample t-Test

One random sample of interval/ratio variable

Comparison with population

or hypothesized mean

Can be either a

one-tailed or a

two-tailed test

df = n-1

n

s

xt

2

Degrees of Freedom

Always related to the sample size

Symbol: df

Differs across statistical tests

In t-tests, Degrees of Freedom depends on the

number of means that are computed:

One sample mean: df = N – 1

Two sample test: df = N – 2

Repeated measures test computes the mean of the

change score (one mean) so df = N – 1

Sample Problem

A group of telemarketers averaged 80 sales per day.

A sample of n=16 people were randomly chosen for

training on a new technique. After training, the

sample averages 85 sales per day, with SS=60.

Does the sample provide sufficient evidence to

conclude that the training leads to higher levels of

sales?

THINK FIRST

We cannot draw the population distribution,

because we do not know its standard deviation.

We can draw the distribution of sales scores

(the scores of the 16 people) because we know the

mean and standard deviation of the sample.

So: draw the curve representing the sample of

sales scores after training.

What range contains about 95% of scores?

Does it include the pre-training mean of 80?

Hypothesis Test: t-Test

Step 1: Define Hypotheses

H0:_______________________________________

HA: _______________________________________

Step 2: Set Criteria for a decision

Alpha = .05 in two-tails combined

Compute df

Use Table or Calculator to find Critical Value of t

Critical Value is t = __________________________

Define Critical Region (draw yourself a picture)

t-Test Continued

Step 3: Collect data and compute test statistics

Means are given to us: M=85, µ=80

Need to compute s2 in order to compute sM

s2 = _______________

sM = _______________

Compute t = (M-µ) / sM

t = ________________

Step 4: Make a decision based on criteria

Use your picture of the t-curve

Effect Size: Cohen’s d

Effect size has the same meaning for the t-Test that it

did with the z-Test

Equation for Cohen’s d used the population standard

deviation: now we do not have that.

Substitute the sample standard deviation

Compute Cohen’s d for this problem

df

SS

M

ndev' std sample

differencemean d sCohen'

Effect size: r2

Another way of measuring effect size

Out of the total variability, how much is accounted for by being in the treatment group?

SS = Σ(X - M)2 = total variability

Variability without treatment effect

Subtract treatment effect (M-µ) from each score

Then subtract Mean, square the difference, and sum

Compute r2 for this problem

22

2

tr

t df

How to Write Results

1. First sentence: What question was tested? Can

be a statement or a question.

2. Procedure / sample: How was the test done?

Who was tested?

3. What were the results? Report sample statistics

(M and sM), the test statistic (t) and its df,

the significance (p), and decision.

4. If significant, report effect size

5. Close with a summary sentence

Sample Narrative

A sample of 16 telemarketers received special

training. After training, their average daily sales

(M = 85, s = 2) were significantly higher than the

pre-training average of 80 for all the telemarketers

(t(15) = 10, p < .05). The effect size, measured

by Cohen’s d, is quite large (d = 2.5).

Approximately 87% of the variability is accounted

for by the training (r2 = .8695). The sales training

produced a significant and sizable difference,

leading to an increase in sales for the firm.

Types of t - Tests

One sample t-Test

Sample mean compared to hypothetical mean or μ

df = 1 because one sample mean is computed

Independent samples t-Test

Two samples are compared to each other

df = 2 because two sample means are computed

Correlated/Paired/Repeated Measures t-test

Two related measures; mean difference is computed

df = 1 because one mean is computed

Basic dynamic of all of our tests

They will involve a ratio (a fraction)

The numerator (top) will measure variability

between group(s)

The denominator (bottom) will measure the

variability that is due to random chance

“Difference on the top, and error on the bottom”

Statz Rappers