Hypothesis Testing with Z Tests: Comparing Scores and Percentiles, Lecture notes of Business Statistics

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HYPOTHESIS TESTINGHYPOTHESIS TESTING WITH

Z^ TESTS

Arlo Clark-Foos

Review: Standardization

^ Allows us to easily see how one score (or sample) ^ Allows us to easily see how one score (or sample)compares with all other scores (or a population).

CDC Example: Jessica

p

^ 1. Percentile: How many 15 year old girls are ^ 1. Percentile: How many 15 year old girls areshorter than Jessica?^ †^ 50% + 33.65% = 83.65%

CDC Example: Jessica

p

^ 2. What percentage of 15 year old girls are taller ^ 2. What percentage of 15 year old girls are tallerthan Jessica?^ †^ 50% - 33.65%

OR

100% - 83.65% = 16.35%

CDC Example: Manuel

p

^ Manuel is 15 years old and 61.2 in. tall ^ Manuel is 15 years old and 61.2 in. tall ^ For 15 year old boys,

μ^ = 67,

σ^ = 3.

(^

)^

(^82). 1 (^19). 3 ) 67 (^2). (^61) (

−= − = − = σ

μ X z

^ Consult z table for 1.

Æ^ 46.56%

CDC Example: Manuel

p

^ 1. Percentile ^ 1. Percentile^ †^ Negative z, below mean: 50% - 46.56% = 3.44%

CDC Example: Manuel

p

^ 3. Percent as extreme as Manuel ^ 3. Percent as extreme as Manuel^ †^ 3.44% + 3.44% = 6.88%

Percentages to z Scores

g

^ SAT Example:

μ^ = 500,

σ^ = 100

^ SAT Example:

μ^

σ^

^ You find out you are at 63

rd^ percentile

^ Consult z table for 13%

Æ^ z

=^33

^ Consult z table for 13%

Æ^ z

X^ = .33(100) + 500 = 533

UMD & GRE Example

p

(^

)^

568 (^

− M^ μ(

)^

568 (^

M M

M

z

^ Consult

z^ table for

z^ = 1.

Æ^

^ 50% + 40

Assumptions of Hypothesis Testing

p^

yp^

g

1 The DV is measured on an interval scale1.^ The DV is measured on an interval scale2.^ Participants are randomly selected 3 The distribution of the population is approximately3.^ The distribution of the population is approximatelynormalRobust:

These

hyp^

tests are those that produce fairly

Robust:

These

hyp. tests are those that produce fairly accurate results even when the data suggest that thepopulation might not meet some of the assumptions.p p^

g^

p

†^ Parametric Tests †^ Nonparametric Tests

Testing Hypotheses (6 Steps)

g^

yp^

(^

p )

1 Identify the population, comparison distribution,1.^ Identify the population, comparison distribution,inferential test, and assumptions 2 State the null and research hypotheses2.^ State the null and research hypotheses3.^ Determine characteristics of the comparisondistributiondistribution^ †^

Whether this is the whole population or a controlgroup, we need to find the mean and some measureof spread (variability).

Testing Hypotheses (6 Steps)

g^

yp^

(^

p )

4 Determine critical values or cutoffs4.^ Determine critical values or cutoffs^ †^

How extreme must our data be to reject the null? † Critical Values:

Test statistic values beyond which we

†^ Critical Values:

Test statistic values beyond which we will reject the null hypothesis (cutoffs) „ p levels (

α ):^ Probabilities used to determine the critical value

5.^ Calculate test statistic (e.g., z statistic)6.^ Make a decision^ †^

Statistically Significant:

Instructs us to reject the null

hypothesis because the pattern in the data differs fromh^

ld^

b^ h^

l

what we would expect by chance alone.

The z Test: An Example

p

2 State the null and research hypotheses2.^ State the null and research hypotheses

H^ :^ μ^0

≤^ μ 1 2 H^ :^ μ

^ μ H:^ μ^1

^ μ 1 2 OROR H : μ =^0^1

μ^2 H^ :^ μ^1

≠^ μ 1 2

The z Test: An Example

p

3 Determine characteristics of comparison3.^ Determine characteristics of comparisondistribution.^ †^

Population:

μ^ = 156.5,

σ^ = 14.

p^

μ^

,

†^ Sample:

M^ = 156.11,

N = 97

σ= (^) σ MN

N