document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184272>Question 1184272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A health officer is trying to study the malaria situation of Zambia. From the records of<BR>\n" );
document.write( "seasonal blood survey (SBS) results he came to understand that the proportion of people<BR>\n" );
document.write( "having malaria in Zambia was 3.8% in 2015. The size of the sample considered was 15,000.<BR>\n" );
document.write( "He also realized that during the year that followed (2016), blood samples were taken from<BR>\n" );
document.write( "10,000 randomly selected persons. The result of the 2016 seasonal blood survey showed<BR>\n" );
document.write( "that 200 persons were positive for malaria. Help the Health officer in testing the hypothesis<BR>\n" );
document.write( "that the malaria situation of 2016 did not show any significant difference from that of 2015<BR>\n" );
document.write( "(take the 5% level of significance). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849862 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=849862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to perform a hypothesis test to determine if the malaria situation in 2016 was significantly different from 2015:\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):** There is no significant difference in the proportion of people with malaria between 2015 and 2016.  (p1 = p2)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a significant difference in the proportion of people with malaria between 2015 and 2016. (p1 ≠ p2)\r<BR>\n" );
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document.write( "**2. Define the Significance Level:**\r<BR>\n" );
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document.write( "α = 0.05 (5%)\r<BR>\n" );
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document.write( "**3. Calculate the Sample Proportions:**\r<BR>\n" );
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document.write( "* **2015:**<BR>\n" );
document.write( "   - p1 = 0.038 (3.8%)<BR>\n" );
document.write( "   - n1 = 15,000\r<BR>\n" );
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document.write( "* **2016:**<BR>\n" );
document.write( "   - Number of positive cases = 200<BR>\n" );
document.write( "   - p2 = 200 / 10,000 = 0.02 (2%)<BR>\n" );
document.write( "   - n2 = 10,000\r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic (z-score):**\r<BR>\n" );
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document.write( "We'll use a two-proportion z-test.  The formula for the test statistic is:\r<BR>\n" );
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document.write( "z = (p1 - p2) / sqrt[ p(1-p) * (1/n1 + 1/n2) ]\r<BR>\n" );
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document.write( "Where 'p' is the pooled proportion, calculated as:\r<BR>\n" );
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document.write( "p = (x1 + x2) / (n1 + n2) = (0.038 * 15000 + 200) / (15000 + 10000) = (570 + 200) / 25000 = 770 / 25000 = 0.0308\r<BR>\n" );
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document.write( "Now, plug the values into the z-score formula:\r<BR>\n" );
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document.write( "z = (0.038 - 0.02) / sqrt[ 0.0308 * (1 - 0.0308) * (1/15000 + 1/10000) ]<BR>\n" );
document.write( "z = 0.018 / sqrt[ 0.0298 * (0.0000667 + 0.0001) ]<BR>\n" );
document.write( "z = 0.018 / sqrt(0.0298 * 0.0001667)<BR>\n" );
document.write( "z = 0.018 / sqrt(0.000004967)<BR>\n" );
document.write( "z = 0.018 / 0.002229<BR>\n" );
document.write( "z ≈ 8.08\r<BR>\n" );
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document.write( "**5. Determine the Critical Value:**\r<BR>\n" );
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document.write( "Since this is a two-tailed test (H1: p1 ≠ p2) at α = 0.05, we need to find the critical z-values that correspond to the tails of the standard normal distribution.  Look up z = ±1.96.\r<BR>\n" );
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document.write( "**6. Make a Decision:**\r<BR>\n" );
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document.write( "* **Compare the test statistic to the critical values:** Our calculated z-score (8.08) is much larger than the critical value (1.96).\r<BR>\n" );
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document.write( "* **Conclusion:** Because the absolute value of the calculated z-score is greater than the critical value, we reject the null hypothesis.\r<BR>\n" );
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document.write( "**7. Interpret the Results:**\r<BR>\n" );
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document.write( "There is sufficient evidence at the 5% level of significance to conclude that there is a statistically significant difference in the proportion of people with malaria between 2015 and 2016 in Zambia.  The malaria situation in 2016 was significantly different from 2015.<BR>\n" );
document.write( " </SPAN>\n" );
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