document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169144>Question 1169144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 5​%. A​ mutual-fund rating agency randomly selects 24 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be ​3.73%. Is there sufficient evidence to conclude that the fund has moderate risk at the a=0.10 level of​ significance? A normal probability plot indicates that the monthly rates of return are normally distributed.<BR>\n" );
document.write( "o=0.05<BR>\n" );
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document.write( "i need to find x to the second power<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851449 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=851449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $\sigma$ be the true standard deviation of the monthly rate of return for the mutual fund.<BR>\n" );
document.write( "The mutual fund qualifies as having moderate risk if $\sigma < 0.05$ (or 5%).\r<BR>\n" );
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document.write( "We are given a sample of $n = 24$ months, and the sample standard deviation is $s = 0.0373$ (or 3.73%).<BR>\n" );
document.write( "We want to test the hypothesis that the true standard deviation is less than 0.05 at a significance level of $\alpha = 0.10$.\r<BR>\n" );
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document.write( "The null hypothesis is $H_0: \sigma \ge 0.05$.<BR>\n" );
document.write( "The alternative hypothesis is $H_a: \sigma < 0.05$.\r<BR>\n" );
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document.write( "Since the population is assumed to be normally distributed and we are testing a hypothesis about the population standard deviation, we will use the chi-square distribution. The test statistic for the standard deviation is:\r<BR>\n" );
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document.write( "$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$\r<BR>\n" );
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document.write( "where $n$ is the sample size, $s$ is the sample standard deviation, and $\sigma_0$ is the hypothesized standard deviation under the null hypothesis. In this case, we use the upper bound of the null hypothesis, $\sigma_0 = 0.05$.\r<BR>\n" );
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document.write( "Plugging in the values:<BR>\n" );
document.write( "$\chi^2 = \frac{(24-1)(0.0373)^2}{(0.05)^2}$<BR>\n" );
document.write( "$\chi^2 = \frac{23 \times (0.00139129)}{0.0025}$<BR>\n" );
document.write( "$\chi^2 = \frac{0.03200}{0.0025}$<BR>\n" );
document.write( "$\chi^2 = 12.8$\r<BR>\n" );
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document.write( "The degrees of freedom for the chi-square distribution are $df = n - 1 = 24 - 1 = 23$.\r<BR>\n" );
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document.write( "Since this is a left-tailed test ($H_a: \sigma < 0.05$), we need to find the critical value $\chi^2_{\alpha, df}$ such that the area to the left of it is $\alpha = 0.10$ with $df = 23$. We look up the chi-square distribution table or use a statistical calculator for $\chi^2_{0.10, 23}$.\r<BR>\n" );
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document.write( "From the chi-square distribution table, the critical value $\chi^2_{0.10, 23} \approx 15.659$.\r<BR>\n" );
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document.write( "Now we compare the test statistic to the critical value:<BR>\n" );
document.write( "Test statistic $\chi^2 = 12.8$<BR>\n" );
document.write( "Critical value $\chi^2_{0.10, 23} \approx 15.659$\r<BR>\n" );
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document.write( "Since the test statistic ($12.8$) is less than the critical value ($15.659$), we reject the null hypothesis.\r<BR>\n" );
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document.write( "There is sufficient evidence at the $\alpha = 0.10$ level of significance to conclude that the standard deviation of the monthly rate of return is less than 5%. Therefore, there is sufficient evidence to conclude that the fund has moderate risk.\r<BR>\n" );
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document.write( "The question asks to find $x^2$. In the context of the chi-square test, $x^2$ usually refers to the chi-square test statistic.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{12.8}$ </SPAN>\n" );
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