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					    Introduction to the t statistic

• Overview
  – The previous chapter used a z statistic to
    determine whether an observed mean differed
    from the population mean
  – Z = (M - )/M
  – Limitation: necessary to know the population SD
  – a t-statistic can be used in situations in which the
    population SD is not known
     Introduction to the t statistic

• The t statistic
  – The t statistic is similar to the z statistic; however,
    unlike the z statistic, it estimates from the data
    the population variability
  – Recall SS = ∑ (X – sample mean)2
  – Sample variance = s2 = SS/ (n-1)
  – Sample standard deviation = s = √(SS/n-1)
        Or s = √SS/df
       Introduction to the t statistic

• The t statistic
   –   When the population variance is known.
   –   Standard error = σM = σ/n.5 = √ σ2/n
   –   Estimated standard error sM = s/n.5 = √ s2/n
   –   Note: sM is an estimate of the standard error
   –   Used when σ is unknown
   –   Computed from the sample variance
   –   Estimates distance between M (sample mean) and 
     Introduction to the t statistic

• The t statistic
  – The t statistic = t = (M-)/ sM
  – where sM = √ s2/n
• Summary
  – the t statistic used to test hypotheses about an
    unknown  when  is unknown. Structure of t
    similar to z
      Introduction to the t statistic

• Similarity of z and t statistic
   – Z = (M - )/M = (M - )/√ 2 /n

   – t statistic = t = (M-)/ sM = (M-)/ √ s2/n

   – Note: 2 statistics are similar except that s2 is used with t
     and 2 is used with z; thus, the t statistic approximates
     the z statistic to the extent that the sample variance
     approximates the population variance
     Introduction to the t statistic

• Degrees of freedom
  – Degrees of freedom, df = n-1
  – The larger the df the better the sample variance
    approximates the population variance
• t distributions
  – By taking all possible samples of a given size (n)
    from a population it is possible to construct a t
    distribution
     Introduction to the t statistic

• t distributions
  – t distributions are symmetrical just like normal
    distributions
  – As the sample size increases the shape of the t
    distribution becomes more similar to normal
    distribution (see Figure 9.1)
  – As this figure shows, the t distribution is flatter
    and more spread out
  – Mean of t distribution is 0
     Introduction to the t statistic

• t distributions
  – In order to test hypotheses it is necessary to
    determine proportions for different t values, just
    as one does with z distributions
  – Appendix B.2 has t distribution table
  – Note: proportion (1 or 2 tailed) is at top; left-most
    row has df; the body of the table has the t-
    distribution value
      Introduction to the t statistic

• t distributions
   – E.g., study involving 20 people evaluates hypothesis with
     population mean memory performance is affected by a
     memory training program
   – Results find a t-value of 2.86. What proportion of scores
     in t-distribution have scores greater than 2.86
   – Sol’n: df = 20-1 =19. look up value in table. Answer .005
      Introduction to the t statistic

• Hypothesis testing with t distributions
   – Logic is the same as with a z score. There is a known
     population mean. Purpose of study is to determine
     whether a treatment changes population mean
     significantly.
   – Step 1. state null and alternative hypothesis
       Population mean is unchanged as a result of treatment H0
       set alpha level
   – Step 2. Locate critical region in t-distribution
   – Step 3. Collect data and compute test statistic
   – Step 4. Evaluate null hypothesis
     Introduction to the t statistic

• Example
  – Purpose of study is to determine whether birds avoid a
    chamber in which eye-spot patterns have been painted.
    Bird placed in a chamber for 60 minutes. 1 compartment
    has eye-spot pattern and other (plain) compartment does
    not.
  – DV = time spent in plain compartment
      H0  = 30 minutes
      H1  ≠ 30 minutes
  – Alpha = .05 two tailed. why two tailed?
    Introduction to the t statistic

• Example
  – Results: n=16 birds; M = 39 minutes on plain
    side; SS = 540
  – Locate critical region
     Df = 15; alpha = .05, two tailed; t (15) ± 2.131
  – Calculate t
  – t = (M-)/ sM = (M-)/ √ s2/n
     S2 = SS/(n-1) = 540/15 = 36
     √ s2/n = √36/16 =√2.25 = 1.50
    Introduction to the t statistic

• Example
  – t = (M-)/ sM = (M-)/ √ s2/n
     = (39-30)/1.50 = 6.0
  – Make decision
     Reject null hypothesis. Birds spent significantly
      more time in the plain compartment than in the
      eye-spotted compartment
    Introduction to the t statistic

• APA format
  – Report descriptive statistics used M = 39
    minutes, SD = 6
  – Report inferential statistics
  – t(15) = 6.0, p < .05
  – t (df) = t score, significance level
    Introduction to the t statistic

• Importance of looking at your data
  – Always look at your data descriptively before
    looking at it inferentially
  – If you graph your data and the raw scores are
    bunched far away from the population mean you
    can be confident that the sample mean differs
    significantly from the population mean
  – See Figure 9.6
    Introduction to the t statistic

• Assumptions of the t test
  – 1. sample must consist of independent
    observations
  – How to ensure: use random sampling
  – 2. population sampled must be normal
  – This assumption is not critical. That is, data show
    that the t statistic is robust when this assumption
    is violated
 Hypothesis test with 2 independent
              samples
• Overview
  – Previously we have shown that a z statistic could
    be used to determine whether an observed mean
    differed from the population mean
  – Z = (M - )/M
  – Limitation: necessary to know the population SD
 Hypothesis test with 2 independent
              samples
• Overview
  – then we showed that a t-statistic can be used in
    situations in which the population SD is not
    known
  – Limitation: the particular type of t statistic can
    only be used when there is a single sample
  – In psychology and other disciplines one often
    wants to determine whether performance
    between two samples (or groups) differs
 Hypothesis test with 2 independent
              samples
• Overview
  – This chapter will present a test that can be used
    in this situation
  – e.g., compare performance of two different
    training procedures administered to two different
    groups
  – e.g., compare performance of two groups trained
    in the same way but tested in two different ways
 Hypothesis test with 2 independent
              samples
• Note:
  – Procedure in this chapter applies to a situation in
    which the data come from two separate samples
  – Hence this procedure is called an independent-
    measures research design or a between-subjects
    design
  – Subsequently we will discuss a procedure in
    which the comparison is made between two sets
    of data collected from the same sample
 Hypothesis test with 2 independent
              samples
• Independent measures or between-subjects
  design
  – Experimental design that uses a different sample
    for each condition or treatment
 Hypothesis test with 2 independent
              samples
• Goal
  – To determine whether there is a significant
    difference between the means of two different
    populations (or two different treatment
    conditions)
  – Notation: a subscript 1 will designate all
    parameters of population/treatment 1, while a
    subscript 2 will designate parameters of
    population/treatment 2
  Hypothesis test with 2 independent
               samples
• Hypothesis
  – H0 : 1 - 2 = 0 in other words the population means are
    equal
  – H1 : 1 - 2 ≠ 0
• Further notation
  – t test used to test the mean of a single sample will be
    referred to as the single sample t statistic
  – t test used to test the mean difference between two
    independent samples will be called the independent
    measures t statistic
  Hypothesis test with 2 independent
               samples
• General formula for t statistic
  – t = (sample statistic –population parameter)/
    standard error
  – Recall APA uses M to designate sample mean
• Single sample t
  – t = (M - )/standard error
• Independent means t test
  – t = ((M1 – M2) – (1 - 2))/standard error
  Hypothesis test with 2 independent
               samples
• Standard error in t statistic
   – Measures how accurately the sample statistic measures
     the population parameter
   – In the case of a single sample t statistic, the standard
     error sM represents the error between a sample mean M,
     and the population mean 
   – In the two sample t statistic, standard error represents the
     error between the sample mean difference (M1 – M2) and
     the population mean difference (1 - 2)
  Hypothesis test with 2 independent
               samples
• Standard error in t statistic
   – Recall each mean estimate, M1 and M2, has associated
     with it an error, which can be estimated by the standard
     error of the mean,
   – sM = √s2/n (formula for single sample standard error)
   – Note: errors add even though you are subtracting the two
     sample means. This makes sense because there are now
     two sources of error
   – This means that sM1-M2 = √(s12/n1+ s22/n2)
   – Caveat: this formula is appropriate only when n1 = n2
  Hypothesis test with 2 independent
               samples
• Standard error in t statistic
  – When n1 ≠ n2 one must used the pooled estimate
    of the variance
  – Recall s2 = SS/ df
  – Pooled estimate of variance = Sp2 = (SS1 +
    SS2)/(df1 + df2)
  – Note: Sp2 is the average of the 2 variances.
    Hence it must be between the two variance
    estimates
  Hypothesis test with 2 independent
               samples
• Standard error in t statistic
  – Two sample standard error = sM1-M2
  – sM1-M2 = √(sp2/n1+ sp2/n2)


• t statistic for independent measures
   – t = ((M1 – M2) – (1 - 2))/ sM1-M2
• df for the t statistic = df1 + df2 = n1- 1 + n2 -1
  Hypothesis test with 2 independent
               samples
             Sample Pop’n     Std error
             data   parameter
 t single    M               √(s2/n
sample
t indepen.   M1- M2   1 - 2    √(sp2/n1+ sp2/n2)
measures
 Hypothesis test with 2 independent
              samples
• Single sample variance
  – S2 = SS/df


• Two independence sample variance
  – Sp2 = (SS1 + SS2)/(df1 + df2)
  Hypothesis test with 2 independent
               samples
• Hypothesis test with independent measures t
  statistic
   – See related example 10.1
• Purpose: to determine whether method of loci
  instructions improve memory for concrete nouns.
  Design: Between subjects design: group 1 (no
  specific instructions), group 2 (method of loci); n=
  10 participants in each group
  Hypothesis test with 2 independent
               samples
• Hypothesis
  – H0 : 1 - 2 = 0 memory performance does not
    differ between the two treatment conditions
  – H1 : 1 - 2 ≠ 0 imagery makes a difference
• Locate critical region
  – Set alpha = .05
  – Df = df1 + df2 = 10 – 1 + 10 – 1 = 18
  – Look up t (18) value ± 2.101
  Hypothesis test with 2 independent
               samples
• Calculate test statistic
  – Results
  – Group 1 (no imagery): n = 10, M = 19, SS = 40
  – Group 2 (imagery): n = 10, M = 26, SS = 50
  – Sp2 = (SS1 + SS2)/(df1 + df2) = (40 + 50)/ (9+9)
  –      = 90/18 = 5.0
  – sM1-M2 = √(sp2/n1+ sp2/n2) = √(5/10 + 5/10 = 1
   Hypothesis test with 2 independent
                samples
• Calculate test statistic
   – t = ((M1 – M2) – (1 - 2))/ sM1-M2
        = (19 – 26) / 1
        = -7


• Make decision
    – M1 = 19; SD = √SS/df = √40/9 = √4.44 = 2.11
    – M2 = 26; SD = √SS/df = √50/9 = √5.55 = 2. 36
    – t (18) = -7 , p < .05, two tailed. Data suggest that memory
      performance under imagery conditions is higher than under no
      imagery instructions
  Hypothesis test with 2 independent
               samples
• Using descriptive statistics to check plausibility of
  inferential conclusion
   – M1 = 19; SD = √SS/df = √40/9 = √4.44 = 2.11
   – M2 = 26; SD = √SS/df = √50/9 = √5.55 = 2. 36
   What these data are saying is that most of the scores
     associated with Group 1 fall between 17 and 21, whereas
     most of the scores associated with Group 2 fall between
     23-24 and 28-29. Since these distributions barely overlap,
     it is plausible to conclude that the two means differ
     significantly from each other
   Recall in z distribution ± 1 SD contains about two thirds of
     the data
  Hypothesis test with 2 independent
               samples
• Using descriptive statistics to check plausibility of
  inferential conclusion
   – M1 = 19; SD = √SS/df = √40/9 = √4.44 = 2.11
   – M2 = 26; SD = √SS/df = √50/9 = √5.55 = 2. 36
• Note: in order to check plausibility of conclusions it
  is important to take into consideration
   – Magnitude of difference between the means
   – The size of the SD
   – The sample sizes of the two distributions
 Hypothesis test with 2 independent
              samples
• Assumptions
  – Observations are independent
  – Two populations must be normally distributed
  – Two populations must have equal variances
 Hypothesis test with 2 independent
              samples
• Assumptions
  – The first two assumptions were already
    discussed in the one-sample t test
  – The third assumption is referred to as
    homogeneity of variance
  – Although modest violations of homogeneity do
    not affect interpretation of data, more severe
    violations make interpretation difficult
 Hypothesis test with 2 independent
              samples
• Testing for homogeneity of variance
  (Hartley’s F-max statistic)
  – Rationale: if population variances are
    homogeneous, then sample variances should
    also be similar
  – Thus, ratio of the two sample variances should
    be close to 1
 Hypothesis test with 2 independent
              samples
• Testing for homogeneity of variance
  (Hartley’s F-max statistic)
  – Procedure: compute sample variance for each
    sample: recall s2 = SS/ df
  – Select largest and smallest s2 and compute ratio
    F-max = s2 (largest)/ s2 (smallest)
    Note: f-max should be close to 1 if there is
     homogeneity; compare value of F-max to Table B.

				
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posted:7/4/2012
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