next up previous contents
Next: t Summary Up: Module 8: Introduction to Previous: Dependent   Contents

Independent Samples t test

3.1. Independent Samples t test

Independent Samples t Test

Methodological Application

Application Distinctions

A Quick note about notation...

Getting to the Distribution of Differences Between Means (part 1).

Getting to the Distribution of Differences Between Means (part 2).

Getting to the Distribution of Differences Between Means (part 3).

Getting to the Distribution of Differences Between Means (part 4).

\(S_{M1}^2 = \frac{S_p^2}{n_1}\) \(S_{M2}^2 = \frac{S_p^2}{n_2}\)

Finally...the t test.

\(t_{calc} = \frac{\overline{X}_1 - \overline{X}_2}{S_{dif}}\)
http://www.math.unb.ca/~knight/utility/t-table.htm

3.2. Hypothesis Testing Example

NHST Example

Step 1

Table 3: Daily Show Group
\(X_1\) \(\overline{X}_1\) \(X_1 - \overline{X}_1\) \(\left(X_1 - \overline{X}_1\right)^2\)
6 6.75 -0.750 0.563
6 6.75 -0.750 0.563
9 6.75 2.250 5.063
8 6.75 1.250 1.563
4 6.75 -2.275 7.563
6 6.75 -0.750 0.563
7 6.75 0.250 0.063
8 6.75 1.250 1.563
\(54 = \sum{X_1}\) \(SOS_1 = 17.50\)
\(8 = n_1\)


\(S_1^2 = \frac{\sum{\left(X_1 - \overline{X}_1\right)}^2}{n_1 - 1} = \frac{SOS_1}{df_1} = \frac{17.50}{8 - 1} = \frac{17.50}{7} = 2.50\)

Table 4: Factor Show Group
\(X_2\) \(\overline{X}_2\) \(X_2 - \overline{X}_2\) \(\left(X_2 - \overline{X}_2\right)^2\)
5 3.875 1.125 1.266
4 3.875 0.125 0.016
3 3.875 -0.875 0.766
1 3.875 -2.875 8.266
5 3.875 1.125 1.266
6 3.875 2.125 4.516
3 3.875 -0.875 0.766
4 3.875 0.125 0.016
\(31 = \sum{X_2}\) \(SOS_2 = 16.875\)
\(8 = n_2\)


\(S_2^2 = \frac{\sum{\left(X_2 - \overline{X}_2\right)}^2}{n_2 - 1} = \frac{SOS_2}{df_2} = \frac{16.875}{8 - 1} = \frac{16.875}{7} = 2.411\)

Step 2(a)

\(S_p^2 = \left(S_1^2\right)*\frac{df_1}{df_t} + \left(S_2^2\right)*\frac{df_2}{...

...} = \left(2.500\right)*\frac{7}{14} + \left(2.411\right)*\frac{7}{14} = 2.4555\)

Step 2(b)

\(S_{M1}^2 = \frac{S_p^2}{n_1} = \frac{2.46}{8} = 0.3075\)



\(S_{M2}^2 = \frac{S_p^2}{n_2} = \frac{2.46}{8} = 0.3075\)

Step 2(c) and Step 2(d)

\(S_{dif}^2 = S_{M1}^2 + S_{M2}^2 = 0.3075 + 0.3075 = 0.615\)
\(S_{dif} = \sqrt{S_{dif}^2} = \sqrt{.62} = 0.78\)

Step 3

http://www.math.unb.ca/~knight/utility/t-table.htm

Step 4

\(t = \frac{\overline{X}_1 - \overline{X}_2}{S_{dif}} = \frac{6.75 - 3.875}{0.78} = 3.69\)

Step 5

3.3. Effect Size

Calculating Effect Size for two Independent Groups

\(d = \frac{\overline{X}_1 - \overline{X}_2}{S_p} = \frac{6.75 - 3.875}{\sqrt{2.46}} = \frac{2.875}{1.57} = 1.83\)

3.4. Statistical Power

Using Delta (\(\delta\)) for Statistical Power

\(\delta = d * \sqrt{\frac{n}{2}}\)
\(\delta = d * \sqrt{\frac{n}{2}} = 1.83 * \sqrt{\frac{8}{2}} = 1.83 * \sqrt{4} = 1.83 * 2 = 3.66\)
Using Delta \(\delta\) to calculate appropriate sample size

  • The more useful way to use \(\delta\) is for calculating adequate sample size during the planning of the study.
    • First, we need to calculate \(\delta\) for a desired power \(.60\) with a one-tailed test at .05 significance level.
, now we can calculate the sample size for a given effect size \(d = .25\).
\(1.90 = .25 * \sqrt{n/2}\)
\(1.90 / .25 = \sqrt{n/2}\)
\(7.6^2 = n/2\)
\(2*57.76 = n\)
\(115.52 = n\)
  • As you can see, this is exactly as we did for the dependent samples situation. Just remember that the \(n\) refers to the number of each group.
  • An Additional comments on Power

    • t tests with evenly distributed participants have greater power than those where the participants are unevenly distributed.
    • The dependent samples design has greater power (all else being equal, such as sample size, effect size, etc.) than the independent samples design.
    • And, as always, the larger the sample size, the greater the power.

    3.5. \(CI_{95}\)

    Calculating a Confidence Interval

    • Note, we are calculating the interval on the difference between means.
    • Recall there are two parts of a confidence interval, the upper limit (UL) and the lower limit (LL).
    • The general form of the equations for each limit are:
    \(LL = \left(-crit\right)*\left(SE\right) + mean\)
    \(UL = \left(+crit\right)*\left(SE\right) + mean\)
    • In the current situation for the differences between means:
    \(LL = \left(-t_{crit}\right)*\left(S_{dif}\right) + \left(\overline{X}_1 - \ove...

...X}_2\right) = -1.761*0.78 + \left(6.75 - 3.875\right) = -1.374 + 2.875 = 1.501\)
    \(UL = \left(+t_{crit}\right)*\left(S_{dif}\right) + \left(\overline{X}_1 - \ove...

...X}_2\right) = +1.761*0.78 + \left(6.75 - 3.875\right) = +1.374 + 2.875 = 4.249\)
    \(LL = 1.50\)
    \(UL = 4.25\)

    Interpretation of Confidence Interval

    • Recall, we had a significance level of .05 (\(t_{crit}\)), so we conducted a 95% confidence interval (\(CI_{95}\)) on the differences between means.
    • The Lower Limit was 1.50 and the Upper Limit was 4.25.
    • So, if we drew an infinite number of random samples of viewers of each show, 95% of the differences between means would be between 1.50 and 4.25.
      • Remember, the mean of the population of differences between means is fixed (but unknown); while each sample has its own differences between means (samples fluctuate).

    3.6. Summary of Section 3

    Summary of Section 3

    Section 3 covered the following topics:

    • The Independent Samples t Test.
    • An NHST example of the Independent Samples t Test.
    • Cohen's d Effect Size
    • Use of delta (\(\delta\)) for Statistical Power
    • Use of delta (\(\delta\)) for calculating a-priori sample size
    • Calculation of Confidence Intervals.


    next up previous contents
    Next: t Summary Up: Module 8: Introduction to Previous: Dependent   Contents
    jds0282 2010-10-15