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 H 0 Z-score one example of an inferential statistic  Must 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 = s 2 = SS / n-1 = SS / df  Sample SD = s = √SS / n-1 = √SS / df Therefore:  Estimated SEM = s M = s / √n = √s 2 / n

The t Statistic Calculation of the single-sample t-test: Formula similar  Z-score however  SEM (  M )  estimated SEM (s M ): t = M - µ / s M t = M - µ / √s 2 / n Degrees of Freedom:  Similar to Z-score, only n-1 values are free to vary  As sample size increases: Estimated SEM (s M )  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 = µ M This is also true for the t distribution.

The t Distribution The shape of the t distribution:  Changes as df changes  A “family” of t distributions exists  Distribution more normal as df increases (Figure 9.1, p 284) Characteristics of a t distribution:  Symmetrical and bell-shaped  µ = 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 (s M ) 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 animals  Moths have developed large eye-spot patterns to ward off predators  Researchers 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 H 0 : µ plainside = 30 minutes H 1 : µ plainside ≠ 30 minutes

1 st Column: df = n – 1 1 st Row: Proportion located in either tail 2 nd Row: Proportion located in both tails Body: The critical t-values specific to df and alpha 1 st Column: df = 16 – 1 = 15 1 st Row: Ignore 2 nd Row: 0.05 (alpha) Body: ?

What would the distribution look like if df were larger? Step 3: Calculate Statistic Variance (s 2 ) s 2 = SS/df s 2 = 540/15 s 2 = 36 Step 3: Calculate Statistic SEM (s M ) s M = √s 2 / n s M = √36 / 16 = √2.25 s M = 1.50 Step 3: Calculate Statistic t statistic t = M - µ / s M t = 39 – 30 / 1.5 = 9 / 1.5 t = 6.0 Step 4: Make a Decision t = 6.0 >  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?

Step 3: Calculate Statistic Same as last example t = 6.0 Step 4: Make Decision t = 6.0 > 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 0.10  95% = 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 statistic  Degrees 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