document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173255>Question 1173255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a. A chi-square goodness of fit test, a chi-square independence test, or a chi-square homogeneity test is always right-tailed. Why?\r<BR>\n" );
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document.write( "b. Suppose you are conducting chi-square goodness of fit test. If the sum of the expected frequencies does not equal the sample size, what do you conclude?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850746 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down these chi-square concepts:\r<BR>\n" );
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document.write( "**a. Why Chi-Square Tests Are Always Right-Tailed**\r<BR>\n" );
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document.write( "* **Chi-Square Statistic:** The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies.<BR>\n" );
document.write( "    * The formula is: χ² = Σ [(O - E)² / E]<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * O = Observed frequency<BR>\n" );
document.write( "        * E = Expected frequency<BR>\n" );
document.write( "* **Squared Differences:** Notice that the numerator of the formula involves squaring the differences (O - E)². Squaring any number, whether positive or negative, always results in a non-negative value.<BR>\n" );
document.write( "* **Non-Negative Statistic:** Since all the terms in the summation are non-negative, the chi-square statistic itself is always non-negative.<BR>\n" );
document.write( "* **Right-Skewed Distribution:** The chi-square distribution is a right-skewed distribution. This means that it has a long tail extending towards the right.<BR>\n" );
document.write( "* **Rejection Region:** Large values of the chi-square statistic indicate a significant difference between the observed and expected frequencies. Therefore, the rejection region for chi-square tests is always in the right tail of the distribution.<BR>\n" );
document.write( "* **Focus on Discrepancies:** The purpose of the chi-square test is to determine if there are significant discrepancies between observed and expected frequencies. Larger discrepancies lead to larger chi-square values, which fall into the right tail of the distribution.\r<BR>\n" );
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document.write( "In essence, because the chi-square statistic is always non-negative and we're interested in detecting large deviations from expected values, the test is always right-tailed.\r<BR>\n" );
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document.write( "**b. Chi-Square Goodness of Fit Test and Unequal Sums**\r<BR>\n" );
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document.write( "* **Expected Frequencies:** In a chi-square goodness of fit test, the expected frequencies represent the frequencies you would expect to see if the null hypothesis were true. They are calculated based on a theoretical distribution or a hypothesized proportion.<BR>\n" );
document.write( "* **Sample Size:** The sample size is the total number of observations in your data.<BR>\n" );
document.write( "* **Sum of Expected Frequencies:** The sum of the expected frequencies should always equal the sample size.<BR>\n" );
document.write( "* **Discrepancy:** If the sum of the expected frequencies does not equal the sample size, it indicates an error in your calculations or in the setup of your test.<BR>\n" );
document.write( "* **Conclusion:**<BR>\n" );
document.write( "    * You have made a mistake in calculating the expected frequencies.<BR>\n" );
document.write( "    * There is a mistake in the data.<BR>\n" );
document.write( "    * The test has been setup incorrectly.<BR>\n" );
document.write( "    * You cannot proceed with the chi-square goodness of fit test until you identify and correct the error.\r<BR>\n" );
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document.write( "It is a basic requirement that the total of the expected frequencies match the total of the observed frequencies (the sample size). If they don't, the test is invalid.<BR>\n" );
document.write( " </SPAN>\n" );
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