CFA CFA Level 1 Test of Independence using Contingency data table

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Test of Independence using Contingency data table

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      Hi,

      I have a question on this Exhibit 30 example. I am not sure why this is a one-tailed test.

      Given Hypothesis set up is H0: R = 0 ; Ha: R != 0

      If the null hypothesis is rejected, then the Alt Hypothesis can be positively correlated and negative correlated. Isn’t this is indicating a two- tailed  test?

      Thanks
      <div class=”clearfix cfa-exhibit-header”>

      Exhibit 30– Test of Independence of Size and Investment Type for 1,594 ETFs

      </div>
      <div id=”CFA0129-R-EXH30″ class=”cfa-exhibit”>

      Step 1 State the hypotheses. H0: ETF size and investment type are not related, so these classifications are independent;
      H: ETF size and investment type are related, so these classifications are not independent.
      Step 2 Identify the appropriate test statistic. <span id=”CFA0129-R-eq101″><span id=”MathJax-Element-101-Frame” class=”MathJax” style=”margin: 0px; padding: 0px; border: 0px; vertical-align: baseline; font-family: var(–content-font); font-size: 16px; color: var(–content-dark); display: inline; font-style: normal; font-weight: normal; line-height: 1.4; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;” tabindex=”0″ role=”presentation” data-mathml=”

      χ2=i=1m(OijEij)2Eij

      “><span id=”m101″ class=”math”><span id=”MathJax-Span-2485″ class=”mrow”><span id=”MathJax-Span-2486″ class=”mrow”><span id=”MathJax-Span-2487″ class=”msup”><span id=”MathJax-Span-2488″ class=”mi”>χ</span><span id=”MathJax-Span-2489″ class=”mn”>2</span></span><span id=”MathJax-Span-2490″ class=”mo”>=</span><span id=”MathJax-Span-2491″ class=”mstyle”><span id=”MathJax-Span-2492″ class=”mrow”><span id=”MathJax-Span-2493″ class=”munderover”><span id=”MathJax-Span-2494″ class=”mo”>∑</span><span id=”MathJax-Span-2495″ class=”mrow”><span id=”MathJax-Span-2496″ class=”mi”>i</span><span id=”MathJax-Span-2497″ class=”mo”>=</span><span id=”MathJax-Span-2498″ class=”mn”>1</span></span><span id=”MathJax-Span-2499″ class=”mi”>m</span></span><span id=”MathJax-Span-2500″ class=”mrow”><span id=”MathJax-Span-2501″ class=”mfrac”><span id=”MathJax-Span-2502″ class=”mrow”><span id=”MathJax-Span-2503″ class=”msup”><span id=”MathJax-Span-2504″ class=”mrow”><span id=”MathJax-Span-2505″ class=”mrow”><span id=”MathJax-Span-2506″ class=”mo”>(</span><span id=”MathJax-Span-2507″ class=”mrow”><span id=”MathJax-Span-2508″ class=”msub”><span id=”MathJax-Span-2509″ class=”mi”>O</span><span id=”MathJax-Span-2510″ class=”mrow”><span id=”MathJax-Span-2511″ class=”mi”>i</span><span id=”MathJax-Span-2512″ class=”mi”>j</span></span></span><span id=”MathJax-Span-2513″ class=”mo”>−</span><span id=”MathJax-Span-2514″ class=”msub”><span id=”MathJax-Span-2515″ class=”mi”>E</span><span id=”MathJax-Span-2516″ class=”mrow”><span id=”MathJax-Span-2517″ class=”mi”>i</span><span id=”MathJax-Span-2518″ class=”mi”>j</span></span></span></span><span id=”MathJax-Span-2519″ class=”mo”>)</span></span></span><span id=”MathJax-Span-2520″ class=”mn”>2</span></span></span><span id=”MathJax-Span-2521″ class=”mrow”><span id=”MathJax-Span-2522″ class=”msub”><span id=”MathJax-Span-2523″ class=”mi”>E</span><span id=”MathJax-Span-2524″ class=”mrow”><span id=”MathJax-Span-2525″ class=”mi”>i</span><span id=”MathJax-Span-2526″ class=”mi”>j</span></span></span></span></span></span></span></span></span></span></span><span class=”MJX_Assistive_MathML” role=”presentation”>

      χ2=i=1m(OijEij)2Eij

      </span></span></span>

      Step 3 Specify the level of significance. 5%
      Step 4 State the decision rule. With (3 − 1) × (3 − 1) = 4 degrees of freedom and a one-sided test with a 5% level of significance, the critical value is 9.4877.

      </div>

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