9.6: Two Independent Samples Exercises

Last updated
Save as PDF

Page ID: 22096

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Exercise \(\PageIndex{1}\)

Determine whether to reject or fail to reject the null hypothesis in the following situations:

\(t(40) = 2.49\), p = 0.01
\(\overline{X_{1}}=64, \overline{X_{2}}=54, n_{1}=14, n_{2}=12, s_{\overline{X_{1}}-\overline{X_{2}}}=9.75, p> 0.05\)

Answer

Reject
Fail to Reject

Exercise \(\PageIndex{2}\)

A researcher wants to know if there is a difference in how busy someone is based on whether that person identifies as an early bird or a night owl. The researcher gathers data from people in each group, coding the data so that higher scores represent higher levels of being busy. Test for a difference between at the .05 level of significance by going through the four steps of Null Hypothesis Significance Testing with data provided in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\)- Descriptive Statistics of Busyness for Two Groups
Sample	Mean	SD	N
Night Owl	19.50	6.14	8
Early Bird	26.67	3.39	9

Answer

Step 1:

Research Hypothesis: Night owls score higher on the busyness survey than early birds. [Your research hypothesis might be opposite, but Dr. MO is a night owl and feels like she is very busy...]
- Symbols: \( \bar{X_{NO}} >\bar{X_{EB}} \)
Null Hypothesis: Night owls score similar on the busyness survey as early birds.
- Symbols: \( \bar{X_{NO}} = \bar{X_{EB}} \)

Step 2: One-tailed test, \(df\) = 15, \(t_{Critical}\) = 1.753

Step 3: \(t_{calculated}\) = -3.03

Step 4: \(t_{critical} < |t_{calculated}|\), Reject the null hypothesis.

Based on our data of early birds and night owls, we can conclude that early birds are busier (\(\overline{X_{EB}}=26.67\)) than night owls (\(\overline{X_{NO}}=19.50\)), \(t(15) = -3.03\), \(p < .05\). Although the means are statistically different, this does NOT support the research hypothesis because Dr. MO thought that Night Owls would be busier.