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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

\(\delta = d * \sqrt{\frac{n}{2}}\)
\(\delta = d * \sqrt{\frac{n}{2}} = 1.83 * \sqrt{\frac{8}{2}} = 1.83 * \sqrt{4} = 1.83 * 2 = 3.66\)
http://www.unt.edu/rss/class/Jon/ISSS_SC/Module008/delta

Using Delta \(\delta\) to calculate appropriate sample size

http://www.unt.edu/rss/class/Jon/ISSS_SC/Module008/delta

An Additional comments on Power

3.5. \(CI_{95}\)

Calculating a Confidence Interval

\(LL = \left(-crit\right)*\left(SE\right) + mean\)
\(UL = \left(+crit\right)*\left(SE\right) + mean\)
\(LL = \left(-t_{crit}\right)*\left(S_{dif}\right) + \left(\overline{X}_1 - \ove...
...X}_2\right) = -1.761*0.78 + \left(6.75 - 3.875\right) = -1.374 + 2.875 = 1.501\)
\(UL = \left(+t_{crit}\right)*\left(S_{dif}\right) + \left(\overline{X}_1 - \ove...
...X}_2\right) = +1.761*0.78 + \left(6.75 - 3.875\right) = +1.374 + 2.875 = 4.249\)
\(LL = 1.50\)
\(UL = 4.25\)

Interpretation of Confidence Interval

3.6. Summary of Section 3

Summary of Section 3

Section 3 covered the following topics:


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