document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179576>Question 1179576</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In 2010, Dr. Bob decided to gather research on the type of disorders that present among his patients. His data collection resulted in the following breakdown of patients by disorder: 54.9% Schizophrenia; 21.1% Major Depression; 7.9% Obsessive-Compulsive Disorder; 4.5% Anxiety Disorder; 2.9% Personality Disorder; 8.8% Other. Information was collected from a random sample 0f 300 patients in 2018 to determine whether or not the data has changed significantly. The sample data is given in the table below. At the α = 0.01 level of significance, test the claim that the disorder breakdown of patients at Dr. Bob's hospital has not changed significantly since 2010.\r<BR>\n" );
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document.write( "Type of disorder per patient in sample:\r<BR>\n" );
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document.write( "Disorder	Count<BR>\n" );
document.write( "Schizophrenia	158<BR>\n" );
document.write( "Major Dispression	65<BR>\n" );
document.write( "Obsessive-Compulsive Disorder	29<BR>\n" );
document.write( "Anxiety Disorder	11<BR>\n" );
document.write( "Personality Disorder	14<BR>\n" );
document.write( "Other	23\r<BR>\n" );
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document.write( "Test Statistic:\r<BR>\n" );
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document.write( "Give the P-value:\r<BR>\n" );
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document.write( "Which is the correct result:\r<BR>\n" );
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document.write( "Do not Reject the Null Hypothesis<BR>\n" );
document.write( "Reject the Null Hypothesis\r<BR>\n" );
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document.write( "Which would be the appropriate conclusion?\r<BR>\n" );
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document.write( "There is enough evidence to suggest that the breakdown of patients by disorder has changed significantly since 2010.<BR>\n" );
document.write( "There is not enough evidence to suggest that the breakdown of patients by disorder has changed significantly since 2010. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850213 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It appears you're dealing with a Chi-Square Goodness of Fit test. Here's how to approach the problem:\r<BR>\n" );
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document.write( "**1. State the Hypotheses**\r<BR>\n" );
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document.write( "* Null Hypothesis (H0): The distribution of disorders in 2018 is the same as in 2010.<BR>\n" );
document.write( "* Alternative Hypothesis (H1): The distribution of disorders in 2018 is different from 2010.\r<BR>\n" );
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document.write( "**2. Calculate Expected Frequencies**\r<BR>\n" );
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document.write( "Multiply the percentages from 2010 by the sample size in 2018 (300) to get the expected counts for each disorder:\r<BR>\n" );
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document.write( "| Disorder | 2010 % | Expected Count |<BR>\n" );
document.write( "|---|---|---|<BR>\n" );
document.write( "| Schizophrenia | 54.9% | 164.7 |<BR>\n" );
document.write( "| Major Depression | 21.1% | 63.3 |<BR>\n" );
document.write( "| Obsessive-Compulsive | 7.9% | 23.7 |<BR>\n" );
document.write( "| Anxiety | 4.5% | 13.5 |<BR>\n" );
document.write( "| Personality | 2.9% | 8.7 |<BR>\n" );
document.write( "| Other | 8.8% | 26.4 |\r<BR>\n" );
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document.write( "**3. Calculate the Chi-Square Test Statistic**\r<BR>\n" );
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document.write( "The formula for the Chi-Square test statistic is:\r<BR>\n" );
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document.write( "χ² = Σ [(Observed Count - Expected Count)² / Expected Count]\r<BR>\n" );
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document.write( "| Disorder | Observed | Expected | (O - E)² / E |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| Schizophrenia | 158 | 164.7 | 0.27 |<BR>\n" );
document.write( "| Major Depression | 65 | 63.3 | 0.05 |<BR>\n" );
document.write( "| Obsessive-Compulsive | 29 | 23.7 | 1.18 |<BR>\n" );
document.write( "| Anxiety | 11 | 13.5 | 0.46 |<BR>\n" );
document.write( "| Personality | 14 | 8.7 | 3.22 |<BR>\n" );
document.write( "| Other | 23 | 26.4 | 0.44 |\r<BR>\n" );
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document.write( "χ² = 0.27 + 0.05 + 1.18 + 0.46 + 3.22 + 0.44 = **5.62**\r<BR>\n" );
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document.write( "**4. Determine Degrees of Freedom**\r<BR>\n" );
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document.write( "Degrees of freedom (df) = Number of categories - 1 = 6 - 1 = 5\r<BR>\n" );
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document.write( "**5. Find the P-value**\r<BR>\n" );
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document.write( "Using a Chi-Square distribution table or calculator with df = 5 and χ² = 5.62, you'll find the P-value.  The P-value is approximately 0.346.\r<BR>\n" );
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document.write( "**6. Compare P-value to Significance Level (α)**\r<BR>\n" );
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document.write( "* α = 0.01 (given)<BR>\n" );
document.write( "* P-value (0.346) > α (0.01)\r<BR>\n" );
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document.write( "**7. Make a Decision**\r<BR>\n" );
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document.write( "Since the P-value is greater than α, we **Do Not Reject the Null Hypothesis**.\r<BR>\n" );
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document.write( "**8. Conclusion**\r<BR>\n" );
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document.write( "**There is not enough evidence to suggest that the breakdown of patients by disorder has changed significantly since 2010.**\r<BR>\n" );
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document.write( "**Important Note:** While the observed counts in the 2018 sample might look different from the 2010 percentages, the Chi-Square test tells us that this difference is not statistically significant. The variation could be due to random chance in the sample. </SPAN>\n" );
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