# Hypothesis Tests with the t Statistic by vrd65316

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Hypothesis Tests with the t Statistic
(based on Gravetter & Wallnau)

Review:

Goal of hypothesis test: Is the obtained result
significantly greater than we would expect by chance.

1. Sample mean approximates the population mean.

2. Standard error gives us an idea how well the
sample mean approximates the population mean. It
tells us how much difference between the sample
mean and the population mean we might expect by
chance.

3. We compare the sample mean with the
hypothesized population mean with the z score
test.

Problem: We usually do not know the population
standard deviation.

2

We will use the sample variability when the population
value is not known.

- See formulas in Chapter for the following
Sample variance
Sample standard deviation
Estimated standard error

“Estimated standard error (sM) is used as an estimate
of the real standard error when the population
standard deviation is unknown. It is computed from
the sample variance or sample standard deviation and
provides an estimate of the standard distance between
a sample mean and the population mean” (p. 282).

t = sample mean minus population mean divided by
the standard error

t statistic test hypotheses about an unknown
population mean when the population standard
deviation is unknown.

Degrees of freedom

“The larger the sample [and the greater the value of
df], the better the sample represents the population.”

Using the t distribution table
3

- See t distribution table in the back of the book

Hypothesis tests with the t statistic

With hypothesis testing, we begin with a population
with an unknown mean and unknown variance, and a

Goal: Use a sample from the “treated” population to
determine if the treatment has any effect.

t statistic:

Numerator: Difference between sample data (sample
mean) and population hypothesis (population mean).

Denominator: How much difference is expected by
chance (error).

When the obtained difference between the sample data
and the hypothesis (numerator) is greater than chance
(denominator), we obtain a large t. Then we conclude
data are not consistent with hypothesis, and we reject
H O.

Steps
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1. State hypotheses (HO and HA) and set alpha level.

2. Determine df. Then use t distribution table
(Appendix B) to locate critical region (and get the
critical value).
3. Collect data.
4. Analyze data. Decide to retain or reject HO by
comparing the calculated value with the critical
value.

Se Example 9.1

- n = 16 birds
- Birds can roam in a box with two chambers:
- One chamber has plain walls.
- Other chamber has two large eye-spot patterns.
- Birds tested one at a time in the box, for 60 minutes.
- Amount of time spent in plain chamber was recorded.
- Birds spend an average of 39 minutes in plain side,
SS = 540.
- Did the eye-spot pattern have an effect on behavior?

1. State hypotheses and set alpha.

HO = Population mean, plain side is
equal to 30 min
HA = Population mean plain side is NOT
equal to 30 min
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p = .05

2. Locate critical region.

df = n – 1

16 – 1 = 15

Look at t distribution table.

critical region value: +/- 2.131

3a. Calculate test statistic.

s2 = SS/df = 540/15 = 36

3b. Use sample variance (s2) to compute estimated
standard error.

sM = SQUARE ROOT OF s2/n = SQRT 36/16 =
SQRT 2.25 = 1.50

3c. Calculate t statistic from sample data.

t = sample mean-population mean/standard error
= 39-30/1.50 = 9/1.50 = 6.00
** 6.00 IS THE OBTAIN VALUE, ALSO KNOWN AS
6

THE CALCULATED VALUE

4. Make decision regarding HO.

Measuring effect size for the t statistic

mean difference
Cohen’s d =     _______________________

sample standard deviation

Example 9.2

sample standard deviation
S = SQRT SS/df = SQRT 540/15 = sqrt 36 = 6

Cohen’s d =

39-30/6 = 9.6 = 1.5
** COMPARE 1.5 WITH TABLE 8.2

Sample Variance and Hypothesis Testing

High variance means there are big differences from
one score to another, which makes it difficult to see
any consistent trends or patterns in the data.
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** High variance indicates noise and confusion in the
data and makes it difficult to see what’s going on.

** When sample variance is large, the standard error
also tends to be large. To obtain statistical
significance, you must demonstrate that you results
are greater than chance, where “chance” is measured
by the standard error. When sample variance is large it
is simply more difficult to obtain a statistically
significant outcome.

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