2-tail - H0: μ1 = μ2     H1: μ1 <> μ2

1-tail - H0: μ1 <= μ2    H1: μ1 > μ2

1-tail - H0: μ1 >= μ2    H1: μ1 < μ2

t-test Assumptions

• Observations are independent of each other.
• Insured by SRS and random assignment of subjects to groups.

• Observations are sampled from normal populations.
• Observe histograms for each group.

• Samples come from populations with equal variances.
• Check that ratio of smallest standard deviation to largest is not greater than 3.
• Use t-test for unequal variamces if there is an concern about heterogeneity.

t-test Formulas

Example

Consider an example using the hsb2 dataset with write as the outcome variable and schtyp as the two-group categorical variable.

use http://www.philender.com/courses/data/hsbdemo, clear

ttest write, by(schtyp)

Two-sample t test with equal variances

------------------------------------------------------------------------------
   Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
---------+--------------------------------------------------------------------
  public |     168       52.25    .7549735    9.785575    50.75948    53.74052
 private |      32    55.53125    1.269195     7.17965    52.94271    58.11979
---------+--------------------------------------------------------------------
combined |     200      52.775    .6702372    9.478586    51.45332    54.09668
---------+--------------------------------------------------------------------
    diff |            -3.28125    1.817938               -6.866256     .303756
------------------------------------------------------------------------------
Degrees of freedom: 198

                 Ho: mean(public) - mean(private) = diff = 0

     Ha: diff < 0               Ha: diff ~= 0              Ha: diff > 0
       t =  -1.8049                t =  -1.8049              t =  -1.8049
   P < t =   0.0363          P > |t| =   0.0726          P > t =   0.9637

anova write schtyp

                           Number of obs =     200     R-squared     =  0.0162
                           Root MSE      = 9.42527     Adj R-squared =  0.0112

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |   289.40625     1   289.40625       3.26     0.0726
                         |
                  schtyp |   289.40625     1   289.40625       3.26     0.0726
                         |
                Residual |  17589.4687   198  88.8357008   
              -----------+----------------------------------------------------
                   Total |   17878.875   199   89.843593

regress write i.schtyp

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  1,   198) =    3.26
       Model |   289.40625     1   289.40625           Prob > F      =  0.0726
    Residual |  17589.4688   198  88.8357008           R-squared     =  0.0162
-------------+------------------------------           Adj R-squared =  0.0112
       Total |   17878.875   199   89.843593           Root MSE      =  9.4253

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    2.schtyp |    3.28125   1.817938     1.80   0.073    -.3037561    6.866256
       _cons |      52.25   .7271753    71.85   0.000       50.816      53.684
------------------------------------------------------------------------------

test 2.schtyp

 ( 1)  2.schtyp = 0

       F(  1,   198) =    3.26
            Prob > F =    0.0726

There is also a version of the two-group independent t-test without the assumption that the population variances are equal.

ttest write, by(schtyp) unequal

Two-sample t test with unequal variances

------------------------------------------------------------------------------
   Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
---------+--------------------------------------------------------------------
  public |     168       52.25    .7549735    9.785575    50.75948    53.74052
 private |      32    55.53125    1.269195     7.17965    52.94271    58.11979
---------+--------------------------------------------------------------------
combined |     200      52.775    .6702372    9.478586    51.45332    54.09668
---------+--------------------------------------------------------------------
    diff |            -3.28125    1.476767               -6.240124   -.3223763
------------------------------------------------------------------------------
Satterthwaite's degrees of freedom:  55.5288

                 Ho: mean(public) - mean(private) = diff = 0

     Ha: diff < 0               Ha: diff ~= 0              Ha: diff > 0
       t =  -2.2219                t =  -2.2219              t =  -2.2219
   P < t =   0.0152          P > |t| =   0.0304          P > t =   0.9848

regress write i.schtyp, robust

Linear regression                                      Number of obs =     200
                                                       F(  1,   198) =    5.01
                                                       Prob > F      =  0.0263
                                                       R-squared     =  0.0162
                                                       Root MSE      =  9.4253

------------------------------------------------------------------------------
             |               Robust
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    2.schtyp |    3.28125   1.465809     2.24   0.026       .39065     6.17185
       _cons |      52.25   .7565153    69.07   0.000     50.75814    53.74186
------------------------------------------------------------------------------

Linear Statistical Models Course