document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170393>Question 1170393</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher suspects that the actual prevalence among children and adolescents is higher than the previously reported prevalence of GAD among children and adolescents.  The previously reported prevalence is 3.9% and the researcher conducts a study to test the accuracy of the previous reported prevalence of GAD by recruiting 98 children and adolescents from various pediatrician's offices and tests them for GAD using the DSM-5.  The researcher determines that the prevalence of GAD is 6.1%.  What should the researchers conclusion be for 5% significance level? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851210 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=851210\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to approach this hypothesis test:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Define the Hypotheses:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Null Hypothesis (H0):** The actual prevalence of GAD among children and adolescents is 3.9% (p = 0.039).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The actual prevalence of GAD among children and adolescents is higher than 3.9% (p > 0.039).\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Set the Significance Level:**\r<BR>\n" );
document.write( "\n" );
document.write( "* α = 0.05 (5% significance level)\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Calculate the Sample Proportion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Sample size (n) = 98<BR>\n" );
document.write( "* Number of children with GAD = 98 * 0.061 = 5.978. Since you cant have a fraction of a child, we will round to 6.<BR>\n" );
document.write( "* Sample proportion (p̂) = 6 / 98 ≈ 0.0612\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Perform the Hypothesis Test (One-Proportion Z-Test):**\r<BR>\n" );
document.write( "\n" );
document.write( "* We'll use a one-proportion z-test because we're dealing with proportions.<BR>\n" );
document.write( "* The test statistic (z) is calculated as:\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$\r<BR>\n" );
document.write( "\n" );
document.write( "    Where:<BR>\n" );
document.write( "    * p̂ = sample proportion (0.0612)<BR>\n" );
document.write( "    * p = population proportion (0.039)<BR>\n" );
document.write( "    * n = sample size (98)\r<BR>\n" );
document.write( "\n" );
document.write( "* Now we plug in the values:\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{0.0612 - 0.039}{\sqrt{\frac{0.039(1-0.039)}{98}}}$\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{0.0222}{\sqrt{\frac{0.039(0.961)}{98}}}$\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{0.0222}{\sqrt{\frac{0.03748}{98}}}$\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{0.0222}{\sqrt{0.0003824}}$\r<BR>\n" );
document.write( "\n" );
document.write( "    $z = \frac{0.0222}{0.01955}$\r<BR>\n" );
document.write( "\n" );
document.write( "    $z ≈ 1.135$\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Determine the Critical Value or P-value:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Since this is a right-tailed test (H1: p > 0.039), we need to find the critical z-value for α = 0.05.<BR>\n" );
document.write( "* Using a standard normal distribution table or calculator, the critical z-value is approximately 1.645.<BR>\n" );
document.write( "* Alternativly we can calculate the p value.<BR>\n" );
document.write( "* Using a z value of 1.135, and a right tailed test, the p value is .128.\r<BR>\n" );
document.write( "\n" );
document.write( "**6. Make a Decision:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Critical Value Method:**<BR>\n" );
document.write( "    * Our calculated z-value (1.135) is less than the critical z-value (1.645).<BR>\n" );
document.write( "    * Therefore, we fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "* **P-value method:**<BR>\n" );
document.write( "    * The p value of .128 is larger than the significance level of .05.<BR>\n" );
document.write( "    * Therefore, we fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**7. Draw a Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* At the 5% significance level, there is not enough evidence to support the researcher's claim that the actual prevalence of GAD among children and adolescents is higher than 3.9%.<BR>\n" );
document.write( "* The researcher should conclude that the difference between the sample proportion and the previously reported proportion is not statistically significant.\r<BR>\n" );
document.write( "\n" );
document.write( "**In summary:** The researcher does not have sufficient evidence to reject the previously reported prevalence.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );