Single-Sample T-Test

Quantitative Methods in HPELS

440:210

Agenda

Introduction The t Statistic Hypothesis Tests with Single-Sample t Test Instat Assumptions

Introduction Recall Inferential statistics:

Use the sample to approximate population Answer probability questions about H0

Z-score one example of an inferential statisticMust have information about the population

standard deviation!

Introduction

The Problem with Z-Scores: In most cases, the population standard

deviation is unknown In these cases, an alternative statistic

is required in order to test a hypothesis

Agenda

Introduction The t Statistic Hypothesis Tests with Single-Sample t Test Instat Assumptions

The t Statistic Estimation of the Standard Error (M):

Population SD is unknown SEM must be estimated with information from the sample only

Recall:Sample variance = s2 = SS / n-1 = SS / dfSample SD = s = √SS / n-1 = √SS / df

Therefore:Estimated SEM = sM = s / √n = √s2 / n

The t Statistic Calculation of the single-sample t-test: Formula similar Z-score however

SEM (M) estimated SEM (sM): t = M - µ / sM

t = M - µ / √s2 / n

Degrees of Freedom: Similar to Z-score, only n-1 values are free to vary As sample size increases:

Estimated SEM (sM) more accurate representation of

SEM (M)

t statistic more accurate representation of Z

The t Distribution

Recall Z distribution With infinite samples, the sampling

distribution: Approaches normal distribution µ = µM

This is also true for the t distribution.

The t Distribution The shape of the t distribution:

Changes as df changesA “family” of t distributions existsDistribution more normal as df increases

(Figure 9.1, p 284)

Characteristics of a t distribution:Symmetrical and bell-shapedµ = 0

Normal distribution has less variability than the t distribution

Why?

Z distribution SEM is calculated and is therefore constant

t distribution SEM is estimated and is therefore variable

As df increases:Estimated SEM (sM) resembles SEM (M)

The t Distribution

Agenda

Introduction The t Statistic Hypothesis Tests with Single-Sample t Test Instat Assumptions

Hypothesis Test: Single-Sample t-Test Example 9.1 (p 288) Overview:

Direct eye contact is avoided by many animalsMoths have developed large eye-spot patterns

to ward off predatorsResearchers want to test the effect of exposure

to eye-spot patterns on the behavior of moth-eating birds

Birds are put in a room (60-min) with two chambers, separated by a doorway (Figure 9.4, p 289)

If no effect equal time in each chamber (Figure 9.3, p 287)

Recall General Process:

1. State hypotheses

2. Set criteria for decision making

3. Sample data and calculate statistic

4. Make decision

Hypothesis Test: Single-Sample t-Test

Step 2: Set Criteria for Decision Alpha () = 0.05 Critical value?

Assume:

n = 16

M = 39 minutes

SS = 540

= ? use the t-test

Step 1: State Hypotheses

H0: µplainside = 30 minutes

H1: µplainside ≠ 30 minutes

1st Column: df = n – 1

1st Row: Proportion located in either tail

2nd Row: Proportion located in both tails

Body: The critical t-values specific to df and alpha

1st Column: df = 16 – 1 = 15

1st Row: Ignore

2nd Row: 0.05 (alpha)

Body: ?

15

0.05

2.131

What would the distribution look like if df were larger?

Step 3: Calculate Statistic

Variance (s2)

s2 = SS/df

s2 = 540/15

s2 = 36

Step 3: Calculate Statistic

SEM (sM)

sM = √s2 / n

sM = √36 / 16 = √2.25

sM = 1.50

Step 3: Calculate Statistic

t statistic

t = M - µ / sM

t = 39 – 30 / 1.5 = 9 / 1.5

t = 6.0

Step 4: Make a Decision

t = 6.0 > 2.131 Accept or Reject?

Confirmation of decision

One-Tailed Single-Sample t-Test Example Example 9.4 (p 297) Overview:

Researchers are still interested in the effect of eye-spot patterns on bird behavior

Based on prior knowledge researchers assume birds will spend less time with eye-spot patterns

Therefore a directional (one-tailed test) will be used

Step 2: Set Criteria for Decision Alpha () = 0.05 Critical value?

15

0.05

1.753

Step 3: Calculate Statistic

Same as last example

t = 6.0

Step 4: Make Decision

t = 6.0 > 1.753

Accept or Reject?

Agenda

Introduction The t Statistic Hypothesis Tests with Single-Sample t Test Instat Assumptions

Instat Type data from sample into a column.

Label column appropriately. Choose “Manage” Choose “Column Properties” Choose “Name”

Choose “Statistics”Choose “Simple Models”

Choose “Normal, One Sample”

Layout Menu: Choose “Single Data Column”

Instat

Data Column Menu:Choose variable of interest

Parameter Menu:Choose “Mean (t-interval)”

Confidence Level:90% = alpha 0.1095% = alpha 0.05

Instat

Check “Significance Test” box: Check “Two-Sided” if using non-directional

hypothesis. Enter value from null hypothesis.

What population value are you basing your sample comparison?

Click OK. Interpret the p-value!!!

Reporting t-Test Results How to report the results of a t-test: Information to include:

Value of the t statisticDegrees of freedom (n – 1) p-value

Example:The average IQ of Black Hawk County 6th

graders was significantly greater than 75 (t(100) = 2.55, p = 0.02)

Agenda

Introduction The t Statistic Hypothesis Tests with Single-Sample t

Test Instat Assumptions

Assumptions of Single-Sample t-Test Independent Observations:

Random selection Normal Distribution:

Tenable if the population is normal If unsure about population assume normality if

sample is large (n > 30) If the sample is small and unsure about population

assume normality if the sample is normal Tests are also available

Scale of Measurement Interval or ratio

Violation of Assumptions Nonparametric Version Chi-Square

Goodness of Fit Test (Chapter 17) When to use the Chi-Square Goodness of Fit

Test:Scale of measurement assumption violation:

Nominal or ordinal data

Normality assumption violation: Regardless of scale of measurement

Textbook Assignment

Problems: 3, 11, 23, 27